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EXPERIMENTS


DESCRIPTION


APPLICATIONS

Ball and Urn Experiment

To view this experiment, click here and select “Ball and Urn Experiment” from the list. 

 

SOCR_experiments.html

 

This experiment exemplifies the binomial distribution.  The user determines the parameters of the binomial distribution (n,p,k), and the computer randomly choose balls according to that distribution.  The red ball is considered a “success”, while the green ball is considered the “failure”.  The blue histogram displays the probabilities of choosing all possible numbers of red balls, while the red histogram displays the observed proportions of the cumulative number of red balls chosen per trial.  The theoretical and observed μ and σ are displayed beneath the histograms.  The numerical probabilities and observed proportions of each result are also given. 

Binomial Distribution

BallotExperiment:

To view this experiment, click here and select “BallotExperiment:

” from the list. 

 

SOCR_experiments.html

 

In an election, candidate A receives a votes and candidate B receives b votes, where a > b. The votes are assumed to be randomly ordered. The first graph shows the difference between the number of votes for A and the number of votes for B, as the votes are counted. The event of interest is that A is always ahead of B in the vote count, or equivalently, that the graph is always above the horizontal axis (except of course at the origin). The indicator variable I of this event is recorded in the first table on each update. The density function of I is shown in blue in the second graph and is recorded in the second table. On each update, the empirical density of I is shown as red in the second graph and recorded in the second table. The parameters a and b can be varied with scroll bars.

 

BertrandExperiment:

To view this experiment, click here and select “BertrandExperiment” from the list. 

 

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Bertrand's experiment is to generate a random chord of a circle. In the simulation, one point of the chord is fixed at (1, 0) and the other random point (X, Y) is recorded on each update in the first table. Also recorded are D, the length of the line segment from the center of the circle to the center of the chord, and A, the angle that this line segment makes with the horizontal. Variable I indicates the event that the chord is longer than the length of a side of the inscribed equilateral triangle. The density of I is shown in blue in the distribution graph and and is recorded in the distribution table. On each update, the empirical density of I is shown in red in the distribution graph and is recorded in the distribution table. Three different models can be selected with the list box: the model where the distance D is uniformly distributed, the model where the angle A is uniformly distributed, and the model where the coordinate X is uniformly distributed.

 

BetaCoinExperiment:

To view this experiment, click here and select “BetaCoinExperiment” from the list. 

 

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The experiment is to toss a coin n times, where the probability of heads is p. The probability of heads is modeled with a prior β distribution, having parameters a and b. The prior density and the true probability of heads are shown in blue in the graph on the right. On each update, the number of heads Y is recorded in the first table. On each update, the posterior β density, which has parameters a + Y and b + n - Y is shown in red in graph on the right. Also, the Bayesian estimate of p, U = (a + Y) / (a + b + n) is recorded in the first table on each update. Finally, the second table gives the true distBias and mean square error of U, and on each update gives the empirical distBias and mean square error, based on the all of the runs of the experiment. The parameters n, p, a, and b can be varied with scroll bars.

Bayesian Estimation - Prior/Posterior.

BetaEstimateExperiment:

To view this experiment, click here and select “BetaEstimateExperiment” from the list. 

 

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The experiment is to generate a random sample X1, X2, ..., Xn of size n from the β distribution with parameters a and 1. The distribution density is shown in blue in the graph, and on each update, the sample density is shown in red. On each update, the following statistics are recorded:

U = M / (1 - M) where M = (X1 + X2 + ··· + Xn) / n
V = -n / [ln(X1) + ln(X2) + ··· + ln(Xn)].


Statistics U and V are point estimators of a. As the experiment runs, the empirical bias and mean square error of each estimator are recorded in the second table. The parameters a and n can be varied with scroll bars.

Bias & precision in parameter estimation, using Beta-distribution.

BinomialCoinExperiment:

To view this experiment, click here and select “BinomialCoinExperiment” from the list. 

 

SOCR_experiments.html

 

The experiment consists of tossing n coins, each with probability of heads p. The number of heads X and the proportion of heads M are recorded on each update. Either X or M can be selected with the list box. The discrete probability density function and moments of the selected variable are shown in blue in the distribution graph and are recorded in the distribution table. On each update, the empirical probability density function and moments of the selected variable are shown in red in the distribution graph and are recorded in the distribution table. The parameters n and p can be varied with scroll bars.

Number of heads and proportion of heads in Binomial experiments

BinomialTimelineExperiment:

To view this experiment, click here and select “BinomialTimelineExperiment” from the list. 

 

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The experiment consists of performing n Bernoulli trials, each with probability of success p. The successes are recorded as red dots on a timeline marked from 1 to n. The number of successes X and the proportion of successes M are recorded on each update. Either X or M can be selected with the list box. The discrete probability density function and moments of the selected variable are shown in blue in the distribution graph and are recorded in the distribution table. On each update, the empirical probability density function and moments of the selected variable are shown in red in the distribution graph and are recorded in the distribution table. The parameters n and p can be varied with scroll bars.

Number of heads and proportion of heads in Binomial experiments

BirthdayExperiment:

To view this experiment, click here and select “BirthdayExperiment” from the list. 

 

SOCR_experiments.html

 

The birthday experiment is to draw a random sample of n balls with replacement from a population of m balls, numbered from 1 to N. We are interested in the random variable V that gives the number of distinct values in the sample, and the indicator variable I that indicates a duplicate sample Value. On each update, the sample of balls is shown in the first graph; a ball that duplicates a previously chosen ball is shown in red, and otherwise is shown in green. The parameters m and n can be varied with scroll bars.

Birthday Paradox as an experiment!

BivariateGame:

To view this experiment, click here and select “BivariateGame” from the list. 

SOCR_Games.html

Click in the graph to generate bivariate data. The selected points are shown as red dots in the scatter plot, and the coordinates are recorded in the first table. The second table gives the μ and σ of the x and y coordinates, and the correlation. In the graph, the horizontal cross hair is centered at the mean of the x-data and extends one standard deviation on either side. Similarly, the vertical cross hair is centered at the mean of the y-data and extends one standard deviation on either side. Finally, the least squares regression line is shown in the scatter plot.

Regression, correlation demo

BivariateNormalExperiment:

To view this experiment, click here and select “BivariateNormalExperiment” from the list. 

 

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The experiment generates a point (X, Y) from a bivariate normal distribution with μ (0, 0). The point is shown graphically as a red dot in the scatter plot, and the coordinates are recorded on each update. The distribution regression line is shown in blue in the scatter plot, and the sample regression line is shown in red. The density function and moments of X and of Y are shown in the distribution graphs, and the moments are given in the distribution tables. The last table gives the distribution and sample correlation. The distribution standard deviations and correlations can be varied within scroll bars.

Regression and correlation demo where the data is randomly sampled from N(μ,σ2)

BivariateUniformExperiment:

To view this experiment, click here and select “BivariateUniformExperiment” from the list. 

 

SOCR_experiments.html

 

The experiment generates a point (X, Y) from a uniform distribution on a region of the plane. Any of the following regions can be selected with the list box:

the square: -6 < x < 6, -6 < y <6;
the triangle: -6 < y < x < 6; and
the circle: x2 + y2 < 36.


The random point is shown has a red dot in the scatter plot, and the coordinates are recorded on each update. The scatter plot also shows the distribution regression line in blue and the sample regression line in red. The distribution and sample density and moments for X and for Y are shown in the second and third graphs. The distribution and sample moments are given in the second and third tables. The distribution and sample correlation are given in the last table.

Regression and correlation demo where the data is randomly sampled from Uniform Distribution

BuffonCoinExperiment:

To view this experiment, click here and select “BuffonCoinExperiment” from the list. 

 

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The experiment consists of tossing a coin on a floor with square tiles, and is shown graphically in the first picture box. The center (X, Y) of the coin relative to the center of the square is recorded on each update in the first table, and is shown graphically as a red dot in the scatter plot. Variable I indicates the event that the coin crosses a crack, and is recorded on each update in the first table. The density of I is shown in the last graph in blue and is given in the second table. When the experiment runs, the empirical density of I is shown in the last graph in red and is given in the second table. The parameter of the process is the radius r of the coin, and can be varied with the scroll bar

Calculus Based Probability

BuffonNeedleExperiment:

To view this experiment, click here and select “BuffonNeedleExperiment” from the list. 

 

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Buffon's needle experiment consists of dropping a needle on hardwood floor, with floorboards of width 1. The experiment is shown graphically in the first graph box. The angle X of the needle relative to the floorboard cracks and the distance Y from the center of the needle to the lower crack are recorded in the first table on each update. Each point (X, Y) is shown as a red dot in the scatter plot. The variable I indicates the event that the needle crosses a crack. The density of I is shown in blue in the distribution graph and is recorded in the distribution table. On each update, the empirical density of I is shown in red in the distribution graph and is also recorded in the distribution table. Finally, pi is shown in blue in the last graph and is recorded in the last table, and on each update, Buffon's estimate of pi is shown in red in the last graph and is recorded in the last table.  The parameter is the needle length L, which can be varied with a scroll-bar

Calculus Based Probability

CardExperiment:

To view this experiment, click here and select “CardExperiment” from the list. 

 

SOCR_experiments.html

 

The experiment consists of dealing n cards at random from a standard deck of 52 cards. The denomination Yi and suit Zi of the i'th card are recorded for i = 1, ..., n on each update. The denominations are encoded as follows:

1 (ace), 2-10, 11 (jack), 12 (queen), 13 (king).


The suits are encoded as follows: 0 (clubs), 1 (diamonds), 2 (hearts), 3 (spades). The parameter n can be varied with a scroll bar

Probability

ChiSquareFitExperiment:

To view this experiment, click here and select “ChiSquareFitExperiment” from the list. 

 

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The experiment is to select a random sample from a specified distribution and perform the χ2 goodness of fit test to another specified distribution, at a specified level of significance. The distributions are discrete distributions on {1, 2, 3, 4, 5, 6}, and thus the experiment corresponds to rolling n dice, each governed by the same distribution.  The sampling and test distributions can be specified by clicking on the die buttons (these bring up the die probability dialog box.  The density of the true distribution is shown in blue in the first graph; the density of the test distribution is shown in green. The significance level can be selected from a list box and the sample size n can be varied with a scroll bar.  The test statistic V has (approximately) the χ2 distribution with 5 degrees of freedom. The density of V and the critical values are shown in blue in the second graph. On each update, the sample values are recorded in the first table and the sample density is shown in red in the first graph. The value of the test statistic V is shown in red in the second graph. Note that the null hypothesis is rejected if and only if V falls outside of the critical values. Variable I indicates the event that the null hypothesis is rejected. The empirical density of I, as the experiment runs, is shown in red in the last graph and is given in the last table. The values of V and I are recorded in the middle table on each update

 

ChuckALuckExperiment:

To view this experiment, click here and select “ChuckALuckExperiment” from the list. 

 

SOCR_experiments.html

 

In the game of chuck-a-luck, a gambler chooses an integer from 1 to 6 and then three fair dice are rolled. If exactly k dice show the gambler's number, the payoff is k:1. The three dice are shown in the first picture box on each update. Random variable W gives the gambler's net profit on a $1 bet and is recorded in the first table on each update. The density and moments of W are shown in blue in the second picture box and are recorded in the second table. On each update, the empirical density of W is shown in red in the second picture box and recorded in the second table

 

CoinDieExperiment:

To view this experiment, click here and select “CoinDieExperiment:

” from the list. 

 

SOCR_experiments.html

 

The coin die experiment consists of first tossing a coin, and then rolling a red die if the coin lands head or a green die if the coin lands tails. The value of the coin (1 for heads and 0 for tails) and the score on the die that is rolled are recorded on each update. You can specify the die distribution of the dice by clicking on the die probability buttons; these buttons bring up the die probability dialog box. You can define your own distribution by typing probabilities into the text fields of the dialog box, or you can click on one of the buttons in the dialog box to specify one of the following special distributions: fair, 1-6 flat, 2-5 flat, 3-4 flat, skewed left, or skewed right. the probability of heads p for the coin can be varied with a scroll bar. The density and moments of the die that is rolled are shown in the distribution graph and distribution table

Conditional Probability

CoinSampleExperiment:

To view this experiment, click here and select “CoinSampleExperiment” from the list. 

 

SOCR_experiments.html

 

The experiment consists of tossing n coins, each with probability of heads p. Variable Ij gives the outcome (1 for heads, 0 for tails) for coin j. The results are recorded on each update. The parameters n and p can be varied with scroll bars

Probability, law of large numbers

ConfidenceIntervalExperiment:

To view this experiment, click here and select “ConfidenceIntervalExperiment” from the list. 

 

SOCR_experiments.html

 

An experiment demonstrating the effects of confidence level and sample-size on the size of the constructed confidence interval. This experiment randomly samples from a N(μ=0, σ2=1) distribution and constructs a confidence interval for the mean (μ). The same phenomena are observed when sampling from almost all symmetric unimodal distributions. Note how confidence interval widens as the confidence level increases. This is counterbalanced by the sample-size increase. Also, see the effect of the confidence level, 100(1-α)%, on the frequency of intervals missing the estimated parameter (μ=0, in this case).

Confidence intervals, margins of error.

CouponCollectorExperiment:

To view this experiment, click here and select “CouponCollectorExperiment” from the list. 

 

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The coupon collector experiment consists of sampling with replacement from the population {1, 2, ..., m} until k distinct values are obtained. The first graph gives the counts for each population value that occurred in the sample. Random variable W gives the sample size and is recorded in the first table on each update. The density and moments of W are shown in blue in the second graph and are recorded in the second table. On each update, the empirical density and moments of W are shown in red in the second graph and are recorded in the second table. The parameters m and k can be varied with scroll bars

 

CrapsExperiment:

To view this experiment, click here and select “CrapsExperiment” from the list. 

 

SOCR_experiments.html

 

In the game of craps, the shooter rolls a pair of fair die. The rules are as follows: If the shooter rolls a sum of 7 or 11 on the first roll, she wins. If the shooter rolls a sum of 2, 3, or 12 on the first roll, she loses. If the shooter rolls a sum of 4, 5, 6, 8, 9, or 10, this number becomes the shooter's point. The shooter continues to roll the dice until the sum is either the point (in which case she wins) or 7 (in which case she loses). Any of the following bets can be selected: Pass: this is the bet that the shooter will win, and pays 1:1. Don't Pass: this is the bet that the shooter will lose, except that an initial roll of 12 is excluded (that is, the shooter loses, but the don't pass bettor neither wins nor loses). The bet pays 1:1. Field: this is a bet on a single throw.

It pays 1:1 if 3, 4, 9, 10, or 11 is thrown, 2:1 if 2 or 12 is thrown, and loses otherwise. Craps: this is a bet that 2, 3, or 12 will come up on a single throw, and pays 7:1. Craps 2: this is a bet that 2 will come up on a single throw, and pays 30:1. Craps 3: this is a bet that 3 will come up on a single throw, and pays 15:1. Craps 12: this is a bet that 12 will come up on a single throw, and pays 30:1. Seven: this is a bet that 7 will come up on a single throw, and pays 4:1. Eleven: this is a bet that 11 will come up on a single throw, and pays 15:1. Big 6. This is a bet that 6 will be thrown before 7, and pays 1:1. Big 8: this is a bet that 8 will be thrown before 7, and pays 1:1. Hardway 4: this is a bet that (2, 2) will be thrown before 7 or any other 4, and pays 7:1. Hardway 6. this is a bet that (3, 3) will be thrown before 7 or any other 6, and pays 9:1. Hardway 8: this is a bet that (4, 4) will be thrown before 7 or any other 8, and pays 9:1. Hardway 10: this is a bet that (5, 5) will be thrown before 7 or any other 10, and pays 7:1

Probability

DiceExperiment:

To view this experiment, click here and select “DiceExperiment” from the list. 

 

SOCR_experiments.html

 

The experiment consists of rolling n dice, each governed by the same probability distribution. You can specify the die distribution by clicking on the die probability button; this button bring up the die probability dialog box. You can define your own distribution by typing probabilities into the text fields of the dialog box, or you can click on one of the buttons in the dialog box to specify one of the following special distributions:

fair, 1-6 flat, 2-5 flat, 3-4 flat, skewed left, or skewed right.


The following random variables are recorded on each update: the sum of the scores Y, the average score M, the minimum score U the maximum score V, the number of aces (1's) Z. Any one of these variables can be selected with a list box. The density and moments of the selected variable are shown in blue in the distribution graph and are recorded in the distribution table. When the simulation runs, the empirical density and moments are shown in red in the distribution graph and are recorded in the distribution table. The parameter n can be varied with a scroll bar.

Discrete variables

DiceSampleExperiment:

To view this experiment, click here and select “DiceSampleExperiment” from the list. 

 

SOCR_experiments.html

 

The experiment consists of rolling n dice, each governed by the same probability distribution. You can specify the die distribution by clicking on the die probability button; this button brings up the die probability dialog box. You can define your own distribution by typing probabilities into the text fields of the dialog box, or you can click on one of the buttons in the dialog box to specify one of the following special distributions: fair, 1-6 flat, 2-5 flat, 3-4 flat, skewed left, or skewed right. The number of dice can be varied with a scroll bar. The scores of each die are recorded in the table.

 

DieCoinExperiment:

To view this experiment, click here and select “DieCoinExperiment” from the list. 

 

SOCR_experiments.html

 

The die coin experiment consists of rolling a die and then tossing a coin the number of times shown on the die. The die score X and the number of heads Y are recorded on each update. The density function and moments of Y are shown in blue in the second graph and are recorded in the second table. On each update, the empirical density and moments of Y are shown in red in the second graph and are recorded in the second table.  The probability of heads for the coin can be varied with the scroll bar. You can specify the die distribution by clicking on the die probability button; this button brings up the die probability dialog box.


You can define your own distribution by typing probabilities into the text fields of the dialog box, or you can click on one of the buttons in the dialog box to specify one of the following special distributions: fair, 1-6 flat, 2-5 flat, 3-4 flat, skewed left, or skewed right.

Inversion of conditioning in conditional probability calculations:

P(D3 | 2H) =
P(2H | D3) P(D3) / P(2H)


RHS = difficult;    LHS can be computed exactly (not hard). Validate empirically that the two are approximately equal. Demo SOCR Dice-Coin Experiment
http://socr.stat.ucla.edu/

ErrorGame:

To view this experiment, click here and select “ErrorGame” from the list. 

SOCR_Games.html

Click on the horizontal axis in the first graph to select points for a frequency distribution. The frequency distribution is given in the first table, and the corresponding histogram is shown in the first graph. The class width (and hence the number of classes) can be varied with the scroll bar. The vertical axis can be selected as either frequency or relative frequency from the list box.  The second graph shows the error function, which can be selected as either mean square error or mean absolute error. In the first case, the minimum occurs at the mean μ and the minimum value is the variance σ2.  In the second case, the minimum occurs throughout the median interval and the minimum value is the mean absolute deviation from the median.

 

Exponential Experiment:

To view this experiment, click here and select “Exponential Experiment” from the list. 

 

SOCR_experiments.html

 

This experiment takes random samples from a user defined exponential distribution and plots the observed results (in red) against the theoretical distribution (in blue).  The observed value for each trial is displayed beneath the distribution in the left column. The theoretical and observed μ and σ are displayed beneath the distribution in the right column. 

Exponential Distribution

FiniteOrderStatisticExperiment:

To view this experiment, click here and select “FiniteOrderStatisticExperiment” from the list. 

 

SOCR_experiments.html

 

The experiment consists of selecting n balls at random (without replacement) from a population of N balls, numbered from 1 to N. For i = 1, 2, ..., n, the i'th smallest number in the sample, X(i) (the i'th order statistic), is recorded on each update. For a selected k, the distribution and moments of X(k) are shown in the distribution graph and the distribution table. The parameters N, n, and k can be varied with scroll bars

 

FireExperiment:

To view this experiment, click here and select “FireExperiment” from the list. 

 

SOCR_experiments.html

 

The forest consists of a rectangular lattice of trees. Each tree in the forest
is in one of three possible states: Healthy trees are colored green; on fire trees are colored red; burnt trees are colored black. A tree that is on fire at time t will be burnt at time t + 1; once a tree is burnt it remains burnt; if a tree is healthy at time t and is directly above or below or to the right or to the left of a tree on fire at time t, then the healthy tree will catch on fire at time, t + 1 independently, with respective probabilities pu,  pd, pr, and pl. The forest is shown in the large graph box in the center. You can choose either a100 by 50 forest or a 500 by 250 forest. To change the state of a tree, select the desired state with the list box and then click on the tree. The fire spread probabilities can be varied with scroll bars. The number of trees that are healthy, on fire, and burnt are recorded at each time unit in the table

 

GaltonBoardExperiment:

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SOCR_experiments.html

 

The Galton board experiment consists of performing n Bernoulli trials with probability of success p. The trial outcome are represented graphically as a path in the Galton board: success corresponds to a bounce to the right and failure to a bounce to the left. The number of successes X and the proportion of successes M are recorded on each update. Graphically, X is the number of the peg in the last row of the Galton board that the ball hits. The distribution and moments of the selected variable (X or M) are given in the distribution graph and the distribution table. The parameters n and p can be varied with scroll bars

Quincunx, Galton Board Game

 

GaltonBoardGame:

To view this experiment, click here and select “GaltonBoardGame” from the list. 

SOCR_Games.html

Click the left arrow button to move the ball to the left and the right arrow button to move the ball to the right. The bit string is recorded in the table below, as well as the corresponding subset. Finally, the last table entry gives the current ball location and the corresponding binomial coefficient

Quincunx, Galton Board Game

 

GammaEstimateExperiment:

To view this experiment, click here and select “GammaEstimateExperiment” from the list. 

 

SOCR_experiments.html

 

The experiment is to generate a random sample X1, X2, ..., Xn of size n from the Γ distribution with shape parameter k and scale parameter b. The distribution density is shown in blue in the graph, and on each update, the sample density is shown in red. On each update, the following statistics are recorded:

U = M2 / T2 , V = T2 / M, W = M / k
where M = (X1 + X2 + ··· + Xn) / n and
T2 = [(X1 - M)2 + (X2 - M)2 + ··· + (Xn - M)2] / n


Statistic U is a point estimator of k and statistics V and W are point estimators of b. As the experiment runs, the empirical bias and mean square error of each estimator are recorded in the second table. The parameters k, b, and n can be varied with scroll bars

 

GammaExperiment:

To view this experiment, click here and select “GammaExperiment” from the list. 

 

SOCR_experiments.html

 

The Γ experiment consists of running a Poisson process until the time of the k'th arrival. The arrivals are shown as red dots in the timeline. The density function of the k'th arrival Tk is shown in the graph box, and the moments are given in the distribution table. The arrival times are recorded in the update table. The parameters are the rate of the process r and the arrival number k, which can be varied with scroll bars.

 

General Birthday Experiment

To view this experiment, click here and select “General Birthday Experiment

” from the list. 

 

SOCR_experiments.html

 

This experiment illustrates the birthday paradox.  The experiment makes a histogram plot of the expected proportions (blue) and the observed proportions (red) of the number of people who do not share the same birthday.  The theoretical and observed μ, σ, and proportions of people who do not share the same birthday are displayed underneath the histograms in the right column.  The results of each experiment are the left column. 

Birthday Paradox as an experiment

MarkovChainExperiment:

To view this experiment, click here and select “MarkovChainExperiment” from the list. 

 

SOCR_experiments.html

 

This applet illustrates a basic Markov chain on the states {0, 1, ..., n}. The states are represented as balls, and the current state is colored red. The number of states can be varied, as well as the transition probabilities. The initial state can be specified by clicking on a ball when the time is 0. The graph and table show the proportion of time spent in each state.  As the time increases, these proportions converge to the steady-state distribution  of the chain.

 

MatchExperiment:

To view this experiment, click here and select “MatchExperiment” from the list. 

 

SOCR_experiments.html

 

The matching experiment is to randomly permute n balls, numbered 1 to n. A match occurs whenever the ball number and the position number agree. The matches are shown in red. The number of matches N is recorded on each update. The distribution and moments of N are shown in the distribution graph and the distribution table. The parameter n can be varied with a scroll bar.

Permutations 

MeanEstimateExperiment:

To view this experiment, click here and select “MeanEstimateExperiment” from the list. 

 

SOCR_experiments.html

 

This class models the interval estimation experiment in the standard N(μ, σ2) model. A random sample from N(μ, σ2) is simulated, and then an interval estimate of μ is computed. The user can choose whether to compute the interval based on the standard normal statistic or the student t statistic. The confidence level and sample size can be chosen.

 

MeanTestExperiment:

To view this experiment, click here and select “MeanTestExperiment” from the list. 

 

SOCR_experiments.html

 

The experiment is to select a random sample of size n from a selected distribution and then test a hypothesis about the mean µ at a specified significance level. The distribution can be selected from a list box; the options are the N(μ, σ2), Γ, and uniform distributions. In each case, the appropriate parameters and the sample size n can be varied with scroll bars. The significance level can be selected with a list box, as can the type of test: two-sided, left-sided, or right-sided. The boundary point mo between the null and alternative hypotheses can be varied with a scroll bar. The density of the distribution and µ are shown in blue in the first graph; mo is shown in green. The test can be constructed under the assumption that the σ of the distribution is known or unknown. In the first case, the test statistic has the standard N(μ,σ2) distribution; in the second case the test statistics has the student t distribution with n - 1 degrees of freedom. The density and the critical values of the test statistic are shown in the second graph in blue. On each update, the sample density is shown in red in the first graph and the sample values are recorded in the first table. The sample mean M is shown in red in the first graph and the value of the test statistics (Z or T) is shown in red in the second graph. The variable I indicates the event that the null hypothesis is rejected. On each update, M, Z or T, and I are recorded in the second table. Note that the null hypothesis is reject (I = 1) if and only if the test statistic (Z or T) falls outside of the critical values. Finally, the empirical density of I is shown in red in the last graph and recorded in the last table.

Hypothesis Testing

Mixture EM Experiment

To view this experiment, click here and select “Mixture EM Experiment

” from the list. 

 

SOCR_experiments.html

 

This experiment uses the expectation maximization algorithm to calculate the parameters for the mixture distribution.  The user decides whether or not to use the gauss mix or a linear mix.  After deciding the number of mixtures, clicking the EM run will generate the distributions according to the algorithm.  To observe a different result, clicking the Init Kernels button will generate new mixtures.  Each color on the graph represents a different mixture.  The weight of each mixture is depicted on the graph in the same color as the distribution.  The mean log likelihood is always displayed in red. 

Mixture Distribution, Expectation maximization algorithm. 

MontyHallExperiment:

To view this experiment, click here and select “MontyHallExperiment” from the list. 

 

SOCR_experiments.html

 

In the Monty Hall experiment, there are three doors; a car is behind one and goats are behind the other two. The player selects a door and then the host opens one of the other two doors. The player can then stay with her original selection or switch to the remaining unopened door. The door finally selected by the player is opened and she either wins or loses. Either of two host strategies can be selected with the list box. In the standard strategy, the host always opens a door with a goat); in the blind strategy, the host randomly opens one of the two doors available to him. The player switches doors with probability p, that can be varied with a scroll bar. Variable G indicates the event that the host opened a door with a goat; variable S indicates the event that the player switched doors; and variable W indicates the event that the player won the car. The density of W is shown in the distribution graph and the distribution table

 Probabilities:
W = {Win};
O={first chosen card is the diamond}

1. Never-Swap (stay home) Strategy:
P(W)=P(W|O)P(O)+P(W|Oc)P(Oc)
P(W)= 1 * 1/3 + 0 * 2/3 = 1/3

2. Always-Swapping Strategy:
P(W)=P(W|O)P(O)+P(W|Oc)P(Oc)
P(W)= 0 * 1/3 + 1 * 2/3 = 2/3

MontyHallGame:

To view this experiment, click here and select “MontyHallGame” from the list. 

SOCR_Games.html

In the Monty Hall game, there are three doors; a car is behind one and goats are behind the other two. Click on the Play button to start a new game. Then select a door by clicking. The host will open one of the other two doors. Finally select a door again (you may stick with your original selection or switch). This door is opened and you either win or lose.
Either of two host strategies can be selected with the list box. In the standard strategy, the host always opens a door with a goat); in the blind strategy, the host randomly opens one of the two doors available to him. Variable G indicates the event that the host opened a door with a goat; variable S indicates the event that you switched doors from your first
selection to your second; and variable W indicates the event that you win the car. The empirical distribution of W is shown in the distribution graph and is recorded in the distribution table

Conditional Probabilities:
W = {Win};
O={first chosen card is the diamond}

1. Never Swap Strategy:
P(W)=P(W|O)P(O)+P(W|Oc)P(Oc)
P(W)= 1 * 1/3 + 0 * 2/3 = 1/3

2. Always Swapping Strategy:
P(W)=P(W|O)P(O)+P(W|Oc)P(Oc)
P(W)= 0 * 1/3 + 1 * 2/3 = 2/3

NegativeBinomialExperiment:

To view this experiment, click here and select “NegativeBinomialExperiment” from the list. 

 

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The negative binomial experiment consists of performing Bernoulli trials, with probability of success p, until the k'th success occurs. The successes are shown as red dots in the timeline. The number of trials Y is recorded on each update. The density and moments of Y are shown in the distribution graph and the distribution table. The parameters k and p
can be varied with scroll bars

Binomial distribution

NormalEstimateExperiment:

To view this experiment, click here and select “NormalEstimateExperiment” from the list. 

 

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The experiment is to generate a random sample X1, X2, ..., Xn of size n from the N(μ,σ2) distribution with mean μ and standard deviation σ. The distribution density is shown in blue in the graph, and on each update, the sample density is shown in red. On each update, the following statistics are recorded:

M = (X1 + X2 + ··· + Xn) / n,
S2 = [(X1 - M)2 + (X2 - M)2 + ··· + (Xn - M)2 ] / (n - 1)
T2 = [(X1 - M)2 + (X2 - M)2 + ··· + (Xn - M)2 ] / n
W2 = [(X1 - m)2 + (X2 - m)2 + ··· + (Xn - m)2 ] / n.


The statistic M is a point estimator of the distribution mean μ and the statistics S2, T2, and W2 are point estimators of the distribution variance σ2. As the experiment runs, the empirical bias and mean square error of each estimator are recorded in the second table. The parameters m, d, and n can be varied with scroll bars.

 

OrderStatisticExperiment:

To view this experiment, click here and select “OrderStatisticExperiment” from the list. 

 

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The experiment consists of selecting a random sample of size n from a specified distribution. The sampling distribution can be selected as either the uniform distribution on (0, 1) or the exponential distribution with parameter 1. The first graph shows the density function and moments of the sampling distribution in blue. For each run of the experiment, the sample density and moments are shown in red. The sample values are arranged in increasing order (these are the order statistics) and recorded in first table. For a specified k, the k'th order statistic is recorded on each update in the second table. The density and moments of X(k) are shown in the second graph in blue. As the experiment runs, the empirical density is shown in red. The moments of X(k) are given in the last table. As the experiment runs, the empirical moments are given also. The parameters n and k can be varied with scroll bars.

 

ParetoEstimateExperiment:

To view this experiment, click here and select “ParetoEstimateExperiment” from the list. 

 

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The experiment is to generate a random sample X1, X2, ..., Xn of size n from the Pareto distribution with parameter a. The distribution density is shown in blue in the graph, and on each update, the sample density is shown in red. On each update, the following statistics are recorded:

U = M / (M - 1) where M = (X1 + X2 + ··· + Xn) / n
V = n / [ln(X1) + ln(X2) + ··· + ln(Xn)].


Statistics U and V are point estimators of a. As the experiment runs, the empirical bias and mean square error of each estimator are recorded in the second table. The parameters a and n can be varied with scroll bars.

 

PointExperiment:

To view this experiment, click here and select “PointExperiment” from the list. 

 

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The experiment consists of tossing coins, each with probability of heads p, until either n heads occur or m tails occur. The event that n heads occur before m tails is indicated by the random variable I. On each update, the results of the coin tosses are shown in the first graph. On each update, the number of heads, the number of tails, and the value of I are recorded in the first table. The density of I is shown in blue in the second graph and is recorded in the second table. On each update, the empirical density of I is shown in red in the second graph and is recorded in the second table. The parameters n, m, and p can be varied with scroll bars.

 

Poisson2DExperiment:

To view this experiment, click here and select “Poisson2DExperiment” from the list. 

 

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The experiment is to run a Poisson process in the plane, and record the number of points N in the square [0, w] x [0, w]. The points are shown as red dots in the scatter plot. The number of points N is recorded on each update. The density and moments of N are shown in blue in the distribution graph and are recorded in the distribution table. On each update, the empirical density of N is shown in red in the distribution graph and is recorded in the distribution table. The parameters of the experiment are the rate r of the process and the side length w of the square, which can be varied with scroll bars.

2D Poisson Distribution

PoissonExperiment:

To view this experiment, click here and select “PoissonExperiment” from the list. 

 

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The experiment to run a Poisson process until time t. The arrivals are shown as red dots on a timeline, and the number of arrivals N is recorded on each update. The density and moments of N are shown in the distribution graph and the distribution table. The parameters of the experiment are the rate of the process r and the time t, which can be varied with scroll bars.

1D Poisson Distribution

PoissonSplitExperiment:

To view this experiment, click here and select “PoissonSplitExperiment” from the list. 

 

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The experiment is to run a Poisson process until time t. Each arrival is type 1 with probability p or type 0 with probability 1 - p. In the timeline, the type 1 arrivals are shown as red dots and the type 0 arrivals as green dots. The number of type 1 arrivals M and the number of type 0 arrivals N are recorded on each update. The densities and moments of M and N are shown in the distribution graphs and the distribution tables. The parameters are the rate of the process r, the time t, and the probability p. These can be varied with scroll bars.

 

PokerDiceExperiment:

To view this experiment, click here and select “PokerDiceExperiment” from the list. 

 

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The poker dice experiment consists of rolling 5 fair dice. Random variable V denotes the value of the hand; V = 0: all distinct, V = 1: one pair, V = 2: two pair, V = 3: three of a kind, V = 4: full house, V = 5: four of a kind, V = 6: five of a kind. The die scores  and the value V are recorded on each update in the first table. The density function of V is shown in blue in the graph and recorded in the second table. On each update, the empirical density function of V is shown in red in the graph and recorded in the second table. In the stop frequency list box, you can set the simulation to stop automatically when V is a particular value.

Probability

PokerExperiment:

To view this experiment, click here and select “PokerExperiment” from the list. 

 

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The poker experiment consists of dealing 5 cards at random from a standard deck of 52 cards. V denotes the value of the hand; V = 0: no value, V = 1: one pair, V = 2: two pair V = 3: three of a kind, V = 4: straight, V = 5: flush, V = 6: full house, V = 7: four of a kind, V = 8: straight flush. The value V is recorded on each update in the first table. The density function of V is shown in blue in the graph and recorded in the second table. On each update, the empirical density function of V is shown in red in the graph and recorded in the second table. In the stop frequency list box, you can set the simulation to stop automatically when V is a particular value.

Probability

ProbabilityPlotExperiment:

To view this experiment, click here and select “ProbabilityPlotExperiment” from the list. 

 

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The experiment is to select a random sample of size n from a specified distribution and graphically test the data against a hypothesized parametric family of distributions. The sampling distribution can be chosen from the list box; the options are the N(μ,σ2), exponential, and uniform; in each case the appropriate parameters can be varied with scroll bars. Similarly, the test distribution can be selected from a list box; the options are the standard N(μ,σ2), the uniform distribution on (0, 1), or the exponential distribution with parameter 1. The density function of the sampling distribution is shown in the first graph in blue, and on each update, the sample values are shown in red. The sample values are arranged in increasing order (these are the order statistics) and recorded in the first table on each update. The density function of the test distribution is shown in blue in the second graph. The quantiles of order i / (n + 1) for i = 1, 2, ..., n are shown in red in the second graph and are recorded in the second table. On each update, the order statistic-quantile pairs are plotted in the third graph (this is the probability plot).

 

Problem of Points Experiment

To view this experiment, click here and select “Problem of Points Experiment

” from the list. 

 

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This game illustrates the famous problem of points devised during the renaissance period.  The player n wins when a user specified number of heads appears.  The player m wins when a user specified number of tails wins.  The game continues until one player wins.  The theoretical probability of each player winning is illustrated in the blue histogram, while the observed proportion of wins is depicted with the red histogram.  The results of each trial are displayed in the left half of the green section.  In this half, the first column is the experiment number, the second column is the number of heads rolled, the third column is the number of tails rolled, and the fourth column depicts the winner.  The number “1” indicates that player “n” won, while the number “0” indicates that player “m” won. The theoretical and observed μ, σ, and proportions are displayed in the right green half.

The famous problem of points as an experiment. 

ProportionEstimateExperiment:

To view this experiment, click here and select “ProportionEstimateExperiment” from the list. 

 

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The experiment is to select a random sample of size n from the Bernoulli distribution with parameter p, and then to construct an approximate confidence interval for p at a specified confidence level. The true value of p can be varied with a scroll bar. The density of the Bernoulli distribution and the the value of p are shown in blue in the first graph. The confidence level can be selected from a list box, as can the type of interval--two sided, upper bound, or lower bound. The interval is constructed using quantiles from the standard N(μ,σ2). The standard normal density and the critical values are shown in blue in the second graph. Variables L and R denote the left and right endpoints of the confidence interval and I indicates the event that the confidence interval contains p. The theoretical density of I is shown in blue in the third graph. On each update, the sample density and the confidence interval are shown in red in the first graph, and the computed standard score is shown in red in the second graph. Note that the confidence interval contains p in the first graph if and only if the standard score falls between the critical values in the second graph. The third graph shows the proportion of successes and failures in red. The first table gives the sample values; the second table records L, R, the standard score Z, and I. Finally, the third table gives the theoretical and empirical densities of I.

 

ProportionTestExperiment:

To view this experiment, click here and select “ProportionTestExperiment:” from the list. 

 

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The experiment is to select a random sample of size n from the Bernoulli distribution with parameter p, and then test a hypothesis about p at a specified significance level. The sample size n can be varied with a scroll bar. The significance level can be selected with a list box, as can the type of test: two-sided, left-sided, or right-sided. The boundary point p0 between the null and alternative hypotheses can be varied with a scroll bar. The Bernoulli density and p are shown in blue in the first graph; p0 is shown in green. The test statistic has approximately a standard N(μ,σ2). The standard normal density and the critical values are shown in the second graph in blue. On each update, the sample density is shown in red in the first graph and the sample values are recorded in the first table. The sample proportion M is shown in red in the first graph and the value of the test statistics Z is shown in red in the second graph. The variable I indicates the event that the null hypothesis is rejected. On each update, M, Z, and I are recorded in the second table. Note that the null hypothesis is reject (I = 1) if and only if the test statistic Z falls outside of the critical values. Finally, the empirical density of I is shown in red in the last graph and recorded in the last table.

 

QuantileApplet:

To view this experiment, click here and select “Quantile Applet” from the list. 

 

SOCR_experiments.html

 

This applet displays the value (quantile) x  and the cumulative probability p = F(x) for a variety of distributions. The following distributions can be chosen from the list box:

N(μ,σ2), Γ, χ2, student t, F, beta, Weibull, Pareto, logistic, lognormal,


The quantile x and probability p are shown graphically in terms of either the probability density function or the cumulative distribution function; the type of graph can be chosen from a list box. The parameters of the chosen distribution can be varied with sliders. Enter a new value of x in the quantile field or a new value of p in the probability field and then press Enter to update the display.

Interactive Distributions (THIS APPLET DOES NOT YET WORK)

RandomVariableExperiment:

To view this experiment, click here and select “RandomVariableExperiment” from the list. 

 

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The experiment simulates a value of a random variable with a specified distribution. The value is recorded on each update. The N(μ,σ2), Γ, χ2, student t, F, beta, Weibull, Pareto, logistic, and lognormal distributions can be chosen from the list box. In each case, the parameters of the distribution can be varied with scroll bars. The density and moments of the distribution are shown in the distribution graph, and the moments are given in the distribution table.

 

RandomWalkExperiment:

To view this experiment, click here and select “RandomWalkExperiment” from the list. 

 

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The experiment consists of tossing n fair coins. The position of the random walk after j tosses is the number of heads minus the number of tails. The random variables of interest are the final position, the maximum position, the time of the last zero. The random walk is shown in red in the left graph on each update. The maximum and minimum values are shown as red dots on the right vertical axis; the last zero is shown as a red dot on the horizontal axis. The value of each of the three random variables is recorded in the first table on each update. Any of the three variables can be selected from the list box. The density and moments of the selected variable are shown in blue in the second graph and are recorded in the second table. On each update, the empirical density and moments are shown in red in the second graph and are recorded in the second table.

Probability, Galton's board game

RedBlackExperiment:

To view this experiment, click here and select “RedBlackExperiment:

” from the list. 

 

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In the red and black experiment, a player starts with an initial fortune x and bets (at even stakes) on independent trials for which the probability of winning is p. Play continues until the player is either ruined or reaches a fixed target fortune a.  Either of two player strategies can be selected from a list box. With timid play, the player bets 1 on each trial. With bold play, the player bets her entire fortune or what is needed to reach the target (whichever is smaller). Variable J indicates the event that the player wins (reaches her target) and variable N is the number of trials played. These variables are recorded on each update. The first graph shows the initial and target fortunes and the final outcome. The density of J is shown in the middle graph and table, and the μ of N is shown in the last graph and table. The parameters x and p can be varied with scroll bars, and the target fortune a can be chosen from a list box.

Probability

RedBlackGame:

To view this experiment, click here and select “RedBlackGame” from the list. 

SOCR_Games.html

This class models the red-black game.  A player plays Bernoulli trials against the house at even stakes until she loses her fortune or reaches a specified target fortune.

Probability

RouletteExperiment:

To view this experiment, click here and select “RouletteExperiment:

” from the list. 

 

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The American roulette wheel has 38 slots numbered 00, 0, and 1-36. Slots 00 and 0 are green. Half of the slots numbered 1-36 are red and half are black. The experiment consists of rolling a ball in a groove in the wheel; the ball eventually falls randomly into one of the 38 slots. The roulette wheel is shown in the left graph panel; the ball is shown on each update. One of seven different bets can be selected from the list box: Bet on 1: this is an example of a straight bet, and bet pays 35:1. Bet on 1, 2: this is an example of a split bet, and pays 17:1. Bet on 1, 2, 3: this is an example of a row bet, and bet pays 11:1. Bet on 1, 2, 4, 5: this is an example of 4-number bet, and pays 8:1. Bet on 1-6: this is an example of a 2-row bet, and pays 5:1. Bet on 1-12: this is an example of a 12-number bet, and pays 2:1. Bet on 1-18: this is an example of an 18-number bet, and pays 1:1. On each update, the outcome X is shown graphically in the first panel and recorded numerically in the first table. Random variable W gives the net winnings for the chosen bet; this variable is recorded in the first table on each update. The density function and moments of W are shown in blue in the distribution graph and are recorded in the distribution table. On each update, the empirical density and moments of W are shown in red in the distribution graph and are recorded in the distribution table.

 

SampleMeanExperiment:

To view this experiment, click here and select “SampleMeanExperiment:

” from the list. 

 

SOCR_experiments.html

 

This class models the sample mean experiment. A random sample of a specified size is drawn from a specified distribution. The density of the sampling distribution and of the distribution of the sample mean are shown.

Random Sampling

Sampling Distribution CLT Experiment:

To view this experiment, click here and select “Sampling Distribution CLT Experiment” from the list. 

 

SOCR_Experiments.html

 

This applet is part of the SOCR Experiments: Sampling Distribution (CLT) Experiment, which demonstrates the properties of the sampling distributions of various sample statistics. This applet can be used to demonstrate the Central Limit Theorem (CLT) as well as:
1. Investigate the effect of the Native population on various sample statistics;
2. Study the effects of the sample-sizes on the sampling distribution;
3. Determine which sample statistics may have known distributions (e.g., Normal), under what conditions;
4. Explore the relations between Population parameters and their corresponding sample counterparts;
For example population mean/SD and sample mean/SD, symmetry, distribution shape, etc.

Demonstration of Central Limit Theorem (CLT). See the SOCR General Central Limit Theorem Experiment Activity.

SignTestExperiment:

To view this experiment, click here and select “SignTestExperiment:

” from the list. 

 

SOCR_experiments.html

 

The experiment is to select a random sample of size n from a distribution, and then to perform a hypothesis test about the median m of the distribution at a specified significance level. The distribution can be selected from a list box; the options are the N(μ,σ2), Γ, and uniform distributions. In each case, the appropriate parameters and the sample size n can be varied with scroll bars. The significance level can be selected with a list box. The null hypothesized value m0 of the median can be selected with a scroll bar; the true quantile level p of m0 is also given. The density of the distribution and m are shown in blue in the first graph; m0 is shown in green. The test statistic N is the number of sample values greater than m0. Under the null hypothesis, N has the binomial distribution with parameters n and 1/2. The density of this distribution and the critical values are shown in the second graph in blue. On each update, the sample density is shown in red in the first graph and the sample values are recorded in the first table. The value of the test statistics N is shown in red in the second graph. The variable J indicates the event that the null hypothesis is rejected. On each update, N and J are recorded in the second table. Note that the null hypothesis is reject (J = 1) if and only if the test statistic N falls outside of the critical values. Finally, the empirical density of I is shown in red in the last graph and recorded in the last table.

Hypothesis Test

SpinnerExperiment:

To view this experiment, click here and select “SpinnerExperiment” from the list. 

 

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The experiment consists spinning a spinner and recording the outcome (the sectors are numbered). The number of sectors in the spinner can be varied with the scroll-bar and the probability of each sector (which is proportional to the area of the sector) can be set with the probability dialog. The probability density function and moments of the random variable (the spinner score) are shown in blue in the distribution graph and recorded numerically in the distribution table. When the experiment runs, the empirical density and moments are shown in red in the distribution graph and recorded in the distribution table. In single step mode, the spinner pointer actually spins, and a sound is played whose tone depends on the outcome.

Probability

TriangleExperiment:

To view this experiment, click here and select “TriangleExperiment:

” from the list. 

 

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Two points are chosen at random in the interval [0, 1]; X denotes the first point chosen and Y denotes the second point chosen. Random variable U gives the type of triangle that can be formed from the three sub-intervals: U = 0, the pieces do not form a triangle; U = 1, the pieces form an obtuse triangle; U = 2, the pieces form an acute triangle. The first picture box shows the outcome of the experiment graphically. On each update, the cut points X and Y are shown as red dots, and the triangle is sketched when U = 1 or U = 2. The second picture box shows the sample space and the three events of interest: U = 2 consists of the 2 interior regions; U = 1 consists of the 6 middle regions; U = 0 consists of the outer regions. On each run, (X, Y) is shown as a red dot in the scatter plot, and is recorded in the first table on each update. The probability density function of U is shown in blue in the graph box and is recorded in the graph table. On each update, the empirical density of U is shown in red in the graph box and is recorded in the graph table. Additionally, the value of U is recorded in the first table on each update.

Calculus Based Probability

Two Type Poisson Experiment

To view this experiment, click here and select “Two Type Poisson Experiment

” from the list. 

 

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This experiment depicts a two-type poisson distribution.  The theoretical probabilities of the number of occurrences for each process are depicted with blue histograms, while the observed proportions of the number of occurrences are displayed using red histograms.  The theoretical and observed μ, σ, and proportions of the second process (the graph on the right side of the screen) is illustrated in the right third of the green space.  The theoretical and observed μ, σ, and proportions of the first process (the graph on the left side of the screen) is illustrated in the middle third of the green space.  The results of each experiment are depicted in the left third of the green space.

Two type Poisson Distribution

UniformEstimateExperiment:

To view this experiment, click here and select “UniformEstimateExperiment” from the list. 

 

SOCR_experiments.html

 

The experiment is to generate a random sample X1, X2, ..., Xn of size n from the uniform distribution on (0, a). The distribution density is shown in blue in the graph, and on each update, the sample density is shown in red. On each update, the following statistics are recorded:

U = 2M where M = (X1 + X2 + ··· + Xn) / n
(the sample mean)
V = (n + 1)X(n) / n where X(n) = max{X1, X2, ..., Xn}
(the n'th order statistic)
W = (n + 1)X(1) where X(1) = min{X1, X2, ..., Xn}
(the first order statistic)

 

UrnExperiment:

To view this experiment, click here and select “UrnExperiment” from the list. 

 

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The experiment consists of selecting n balls at random from an urn with N balls, R of which are red and the other N - R green. The number of red balls Y in the sample is recorded on each update. The distribution and moments of Y are shown in blue in the distribution graph and are recorded in the distribution table. On each update, the empirical density and moments of Y are shown in red in the distribution graph and are recorded in the distribution table. Either of two sampling models can be selected with the list box: with replacement and without replacement. The parameters N, R, and n can be varied with scroll bars.

Demos of Hypergeometric and Binomial Distributions. As well as the approximation of hypergeometric by (the simpler) binomial when the sample-size < 5-10% of the population size.

VarianceEstimateExperiment:

To view this experiment, click here and select “VarianceEstimateExperiment” from the list. 

 

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The experiment is to select a random sample of size n from a specified distribution, and then to construct an approximate confidence interval for σ at a specified confidence level. The distribution can be chosen with a list box; the options are N(μ,σ2), Γ, and uniform. In each case, the appropriate parameters and the sample size can be varied with scroll bars. The density, μ, and σ  of the selected distribution are shown in blue in the first graph. The confidence level can be selected from a list box, as can the type of interval--two sided, upper bound, or lower bound. The interval can be constructed assuming either that the distribution mean μ is known or unknown. In the first case the pivot variable V has the χ2 distribution with n degrees of freedom; in the second case the pivot variable V has χ2 distribution with n - 1 degrees of freedom. The density and the critical values of V are shown in blue in the second graph. Variables L and R denote the left and right endpoints of the confidence interval and I indicates the event that the confidence interval contains the distribution mean. The theoretical density of I is shown in blue in the third graph. On each update, the sample density and the confidence interval are shown in red in the first graph, and the value of V is shown in red in the second graph. Note that the confidence interval contains the mean in the first graph if and only if V falls between the critical values in the second graph. The third graph shows the proportion of successes and failures in red. The first table gives the sample values; the second table records L, R, V, and I. Finally, the third table gives the theoretical and empirical densities of I.

 

VarianceTestExperiment:

To view this experiment, click here and select “VarianceTestExperiment” from the list. 

 

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The experiment is to select a random sample of size n from a selected distribution and then test a hypothesis about the standard deviation d at a specified significance level. The distribution can be selected from a list box; the options are the N(μ,σ2), Γ, and uniform distributions. In each case, the appropriate parameters and the sample size n can be varied with scroll bars.  The significance level can be selected with a list box, as can the type of test: two-sided, left-sided, or right-sided. The boundary point d0 between the null and alternative hypotheses can be varied with a scroll bar. The density of the distribution, as well as m and d, are shown in blue in the first graph; d0 is shown in green. The test can be constructed under the assumption that the distribution mean is known or unknown. In the first case, the test statistic has the χ2 distribution with n degrees of freedom; in the second case the test statistics has the χ2 distribution with n - 1 degrees of freedom. The density and the critical values of the test statistic V are shown in the second graph in blue.  On each update, the sample density is shown in red in the first graph and the sample values are recorded in the first table. The sample standard deviation S is shown in red in the first graph and the value of the test statistic V is shown in red in the second graph. The variable I indicates the event that the null hypothesis is rejected. On each update, S2, V, and I are recorded in the second table. Note that the null hypothesis is reject (I = 1) if and only if the test statistic V falls outside of the critical values. Finally, the empirical density of I is shown in red in the last graph and recorded in the last table.

 

VoterExperiment:

To view this experiment, click here and select “VoterExperiment” from the list. 

 

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Each site in the voter array can be in one of 10 states, represented by various colors. At each discrete time unit, a site is chosen at random, and then a neighbor of the site is chosen at random. The state (color) of the site is changed to the state (color) of the neighbor.  The record table shows the number of sites of each color on each update. You can change the state of a site by selecting the desired color in the list box and then clicking on the desired site.