Copyright: 2000, K. Krishnamoorthy
Citation: K. Krishnamoorthy (2000). Statistical Distributions with Applications and StatCalc Software. Etext.net Publisher, Venice, CA


e-book

Statistical Distributions with Applications and StatCalc Software

Etext.net Publisher, Venice, CA

Author:    K.Krishnamoorthy

The book Statistical Distributions with Applications and StatCalc Software can be ordered online at http://www.etext.net.  The book is also available as an e-book in the WinHelp format. The e-book can be purchased as a downloadable file or on CD. For more information contact Etext.net toll-free at 1-888-EEE-TEXT or visit http://www.etext.net/catalog

To view contents of the book click here


 Binomial Distribution

 n = Number of Trials;  k = Number of Successes

To compute probabilities:

Enter the values of the number of trials n, success probability p, and the observed number of successes k; click [image].

Example 1. When n = 20, p = 0.2, and k = 4,

To compute confidence interval:

Example 2.  Suppose that a binomial experiment of 40 trials resulted into 5 successes. To find a 95% confidence interval, enter 40 for n, 5 for the observed number of successes k, and 0.95 for the confidence level; click [image] to get (0.0419, 0.2680).

To compute moments:

Enter the values of n and p; click [image].


 Hypergeometric Distribution

N = Lot Size, M = Number of Defectives, n = Sample Size

where L = max{0, M – N + n}.

To compute probabilities:

Enter the values of the lot size N, number of defectives M in the lot, sample size n, and the observed number of defectives k in the sample; click [image] .

Example 1.  When N = 100, M = 36, n = 20, and k = 3,

To compute confidence intervals:

Example 2. Suppose that inspection of a sample of 20 items from a lot of 100 items revealed 3 defectives. To find a 95% confidence interval for the total number of defectives in the lot, enter 100 for N, 20 for n, 3 for k, and 0.95 for the confidence level; click [image] to get (4, 36). Because of the discreteness of the distribution, the actual coverage probability is 95.2276%.

To compute a one-sided upper limit for the number of defective items in the lot, enter 0.90 for the confidence level and click [image] to get 33.

To compute moments:

Enter the values of the N, M, and n; click [image].


          Poisson Distribution

λ = mean

To compute probabilities:

Enter the values of the mean, and k at which the probability is to be computed; click [image] .

Example 1.    When the mean = 6, k = 5,

and

To compute confidence intervals:

Example 2.    Suppose that a sample 20 observations yielded a total of 140 occurrences. To find a 95% confidence interval for the mean, enter 140 for k, 20 for the sample size, and 0.95 for the confidence level; click [image] to get (5.88853, 8.26027).

To compute moments:

Enter the value of the mean, and click [image].


 Geometric Distribution

k:   Number of Failures Until the First Success

p:   Success Probability

To compute probabilities:

Enter the number of failures k until the first success and the success probability p; click on [image].

Example 1  The probability of observing the first success at the 12th trial, when the success probability is 0.1, can be computed as follows:

Enter  11 for k, and 0.1 for p; click on  [image] to get

and

where X is the number of failures until the first success to occur.

To compute confidence intervals:

Enter the observed number of failures k until the first success and the confidence level; click on click [image].

Example 2   Suppose a binomial experiment took 12 trials until the first success. To find a 95% confidence interval for   p,  enter 11 for k, 0.95 for confidence level; click [image] to get (0.002, 0.285).

To compute moments:

Enter the values of k and p; click on [image].


Negative Binomial

r = Number of Successes;  k = Number of Failures Until the rth Success

To compute probabilities:

Enter the number of successes r, number of failures until the rth success, and the success probability; click [image] .

Example 1.  When r = 20, k = 18, and p = 0.6,

and

To compute confidence intervals:

Suppose that a binomial experiment took 35 failures until the 5th success. To find a 95% confidence interval for the success probability p, enter 5 for r, 35 for k, and 0.95 for the confidence level; click [image] to get (0.04186, 0.24221).

To compute moments:

Enter the values of r and the success probability p; click [image].


Logarithmic Series

where α = −1/[ln(1 - θ)].

To compute probabilities:

Enter the value of θ, and the observed value k;click [image] .

Example 1  When θ = 0.3, k = 3,

and

To compute the MLE of θ:

Enter the sample mean, and click on [image].

Example 2  When the sample mean = 2, the MLE of θ?is 0.715332.


Logistic Variance


Probability
Click this button to compute probabilities


Percentile
Click this button to compute percentiles


Confidence
Click this button to compute confidence interval


Back
Click this button to go to the previous screen


Forward
Click this button for the topic that follows the

current topic


Backward
Click this button for the topic that precedes the

current topic


Moments

Click this button to compute moments


Critical
Click this button to compute the left-tail and right-tail critical points


Contents

Preface

1	Preliminaries

1.1    Random Variables and Expectations

1.2    Moments and Other Functions

1.2.1    Measures of Central Tendency

1.2.2    Moments

1.2.3    Measures of Variability

1.2.4    Measures of Relative Standing

1.2.5    Other Measures

1.2.6    Some Related Functions

1.3    Some Functions Relevant in Reliability Theory

1.4    Model Fitting

1.4.1    The Q-Q Plot

1.4.2    The Chi-square Goodness-of-fit test

1.5    Methods of Estimation

1.5.1    Moment Estimation

1.5.2    Maximum Likelihood Estimation

1.6    Interval Estimation and Hypothesis Testing

1.6.1    Interval Estimation

1.6.2    Hypothesis Testing

1.7    Random Number Generation

1.8    Some Special Functions

Discrete Distributions

2    Binomial Distribution

2.1    Definition

2.2    Moments

2.3    Inferences

2.3.1    Point Estimation

2.3.2    Interval Estimation

2.3.3    Hypothesis Testing

2.4    Properties and Results

2.4.1    Properties

2.4.2    Relation to Other Distributions

2.4.3    Approximations

2.5    Random Number Generation

2.6    Computational Algorithm

2.7    Computing Table Values Using StatCalc

2.8    Examples

3    Hypergeometric Distribution

3.1    Definition

3.2    Moments

3.3    Inferences

3.3.1    Estimation of the Number of Defectives M in a Lot

3.3.2    Estimation of the Lot Size N

3.3.3    Confidence Intervals for M

3.4    Properties and Results

3.4.1    Recurrence Relations

3.4.2    Relation to Other Distributions

3.4.3    Approximations

3.5    Random Number Generation

3.6    Computational Algorithm

3.7    Computing Table Values Using StatCalc

3.8    Examples

4    Poisson Distribution

4.1    Definition

4.2    Moments

4.3    Inferences

4.3.1    Point Estimation

4.3.2    Interval Estimation and Hypothesis Testing

4.3.3    An Exact Confidence Interval for λ1/λ2

4.3.4    Hypothesis Testing for λ1/λ2

4.4    Properties and Results

4.4.1    Properties

4.4.2    Relation to Other Distributions

4.4.3    Approximations

4.5    Random Number Generation

4.6    Computational Algorithm

4.7    Computing Table Values Using StatCalc

4.8    Applications and Examples

5    Geometric Distribution

5.1	Definition

5.2	Moments

5.3	Properties and Results

5.4	Random Number Generation

5.5	Computing Table Values Using StatCalc

6    Negative Binomial  Distribution

6.1    Definition

6.2    Moments

6.3    Inferences

         6.3.1    Point Estimation

         6.3.2    Interval Estimation

         6.3.3    Hypothesis Testing

6.4    Properties and Results

         6.4.1    Properties

         6.4.2    Relation to Other Distributions

6.5    Random Number Generation

6.6    Computational Algorithm

6.7    Computing Table Values Using StatCalc

6.8    Examples

7     Logarithmic Series Distribution

7.1    Definition

7.2    Moments

7.3    Inferences

         7.3.1    Point Estimation

         7.3.2    Interval Estimation

7.4    Properties and Results

         7.4.1    Recurrence Relations

         7.4.2    Properties

7.5    Random Number Generation

7.6    Computational Method

7.7    Computing Table Values Using StatCalc

7.8    Examples

8    Discrete Uniform Distribution

8.1    Definition

8.2    Moments

Continuous Distributions

9     Normal Distribution

9.1    Definition

9.2    Moments

9.3    Inferences

         9.3.1    Point Estimation

         9.3.2    Interval Estimation

         9.3.3    Hypothesis Testing about the Mean

         9.3.4    Hypothesis Testing about the Variance

9.4    Properties and Results

         9.4.1    Properties

         9.4.2    Relation to Other Distributions

9.5    Random Number Generation

9.6    Computational Algorithm

9.7    Computing Table Values Using StatCalc

9.8    Examples

10     Chi-square Distribution

10.1    Definition

10.2    Moments

10.3    Properties and Results

           10.3.1    Properties

           10.3.2    Relation to Other Distributions

           10.3.3    Approximations

10.4    Random Number Generation

10.5    Computational Method

  10.6    Computing Table Values Using StatCalc

  10.7    Applications

11     F Distribution

11.1    Definition

11.2    Moments

11.3    Properties and Results

           11.3.1    Identities

           11.3.2    Relation to Other Distributions

           11.3.3    Series Expansions

           11.3.4    Approximations

11.4    Random Number Generation

11.5    Computational Method

  11.6    Computing Table Values Using StatCalc

  11.7    Applications and Examples

12     Student's t Distribution

12.1    Definition

12.2    Moments

12.3    Properties and Results

           12.3.1    Properties

           12.3.2    Relation to Other Distributions

           12.3.3    Series Expansions

           12.3.4    An Approximation

12.4    Random Number Generation

12.5    Computational Method

  12.6    Computing Table Values Using StatCalc

  12.7    Applications

12.7.1    Inferences about a Normal Mean

12.7.2    Inferences about Two Normal Means

12.7.2    Inferences about a Regression Coefficient

13    Gamma Distribution

13.1    Definition

13.2    Moments

13.3    Inferences

           13.3.1    Maximum Likelihood Estimators

           13.3.2    Moment Estimators

           13.3.3    Interval Estimation

13.4    Properties and Results

           13.4.1    Properties

           13.4.2    Relation to Other Distributions

           13.4.3    An Approximation

13.5    Random Number Generation

13.6    Computational Method

13.7    Computing Table Values Using StatCalc

13.8    Applications and Examples

14     Beta Distribution

14.1    Definition

14.2    Moments

14.3    Inferences

           14.3.1    Moment Estimators

           14.3.2    Maximum Likelihood Estimators

14.4    Properties and Results

           14.4.1    An Identity and Recurrence Relations

           14.4.2    Relation to Other Distributions

14.5    Random Number Generation

14.6    Computational Algorithm

  14.7    Computing Table Values Using StatCalc

  14.8    Applications with an Example

15     Noncentral Chi-square Distribution

15.1    Definition

15.2    Moments

15.3    Properties and Results

           15.3.1    Properties

           15.3.2    Approximations

15.4    Random Number Generation

15.5    Computational Method

15.6    Computing Table Values Using StatCalc

15.7	Applications

16     Noncentral F Distribution

16.1    Definition

16.2    Moments

16.3    Properties and Results

           16.3.1    Properties

           16.3.2    Approximations

16.4    Random Number Generation

16.5    Computational Method

16.6	Computing Table Values Using StatCalc

16.7	Applications

17     Noncentral t Distribution

17.1    Definition

17.2    Moments

17.3    Properties and Results

           17.3.1    Properties

           17.3.2    An Approximation

17.4    Random Number Generation

17.5    Computational Method

17.6    Computing Table Values Using StatCalc

17.7    Applications and Examples

 17.7.1   Power Studies

 17.7.2   Tolerance Factors and Calibration

18     Laplace Distribution

18.1    Definition

18.2    Moments

18.3    Inferences

           18.3.1    Maximum Likelihood Estimators

           18.3.2    Interval Estimation

18.4    Properties and Results

18.5    Random Number Generation

           18.6    Computational Method

18.7    Computing Table Values Using StatCalc

18.8    Applications with an Example

19     Logistic Distribution

19.1    Definition

19.2    Moments

19.3    Maximum Likelihood Estimators

19.4    Properties and Results

19.5    Random Number Generation

19.6    Computational Algorithm

19.7    Computing Table Values Using StatCalc

19.8    Applications

20     Lognormal Distribution

20.1    Definition

20.2    Moments

20.3    Maximum Likelihood Estimators

20.4    Properties and Results

20.5    Random Number Generation

20.6    Computational Method

20.7    Computing Table Values Using StatCalc

20.8    Applications

21     Pareto Distribution

21.1    Definition

21.2    Moments

21.3    Inferences

           21.3.1    Point Estimation

           21.3.2    Interval Estimation

21.4    Properties and Results

21.5    Random Number Generation

21.6    Computational Method

21.7    Computing Table Values Using StatCalc

21.8    Applications

22     Weibull Distribution

22.1    Definition

22.2    Moments

22.3    Point Estimation

22.4    Properties and Results

22.5	    Random Number Generation

22.6	    Computational Method

22.7	    Computing Table Values Using StatCalc

22.8    Applications

23     Extreme Value Distribution

23.1    Definition

23.2    Moments

23.3    Maximum Likelihood Estimators

23.4    Relation to Other Distributions

23.5    Random Number Generation

23.6    Computing Table Values Using StatCalc

23.7    Applications

24     Cauchy Distribution

24.1    Definition

24.2    Moments

24.3    Inferences

         24.3.1    Estimation Based on Sample Quantiles

         24.3.2    Maximum Likelihood Estimation

24.4    Properties

24.5    Random Number Generation

24.6    Computing Table Values Using StatCalc

24.7    Applications

25     Uniform Distribution

25.1     Definition

25.2     Moments

25.3     Inferences

25.4     Properties and Results

25.5     Random Number Generation

Some Nonparametric Distributions

26     Distribution of Runs

26.1    Description and Moments

26.2    Computing Table Values Using StatCalc

26.3    Examples

27     Sign Test and Confidence Interval for a Median

27.1    Hypothesis Testing for the Median

27.2	Confidence Interval for a Median

27.3	Computing Table Values Using StatCalc

27.4    An Example

28     Wilcoxon Signed-Rank Tes

28.1    Description

28.2	Moments and an Approximation

28.3	Computing Table Values Using StatCalc

28.4    An Example

29     Wilcoxon Rank-Sum Test

29.1    Description

29.2    Moments and an Approximation

29.3    Mann-Whitney U Statistic

29.4    Computing Table Values Using StatCalc

29.5    An Example

30     Nonparametric Tolerance Interval

30.1    Description

30.2    Computing Table Values Using StatCalc

30.3    An Example

31     Fisher's Exact Test for 2 x 2 Table

31.1    Description

31.2    Sample Size Determination

31.3    Computing Table Values Using StatCalc

31.4    Examples

Miscellaneous

32     Tolerance Factors for a Normal Population

32.1    Description

32.2    Computing Tolerance Factors Using StatCalc

32.3    Examples

33     Tolerance Factors for a Multivariate Normal Population

33.1    Description

33.2    Computing Tolerance Factors Using StatCalc

33.3    Examples

34     Bivariate Normal Distribution

34.1    Definition

34.2    Some Properties

34.3    Random Number Generation

34.4    Computational Algorithm

34.5    Computing Table Values Using StatCalc

34.6    An Example

35     Distribution of the Sample Correlation Coefficient

35.1    Description

35.2    Inferences

         35.2.1    Point Estimation

         35.2.2    Interval Estimation

         35.2.3    Hypothesis Testing

35.3    Random Number Generation

35.4    Computing Table Values Using StatCalc

35.5    An Example

36     Distribution of the Sample Multiple Correlation Coefficient

36.1    Description

36.2    Moments

36.3    Inferences

           36.3.1    Point Estimation

           36.3.2    Interval Estimation

           36.3.3    Hypothesis Testing

36.4    Some Results

36.5    Random Number Generation

36.6    Computational Method

36.7    Computing Table Values Using StatCalc

37     Distribution of the Maximum of k |t| Variables

37.1    Description

37.2    An Application

37.3    Computing Table Values Using StatCalc

37.4    An Example

References

Index


Normal Distribution

μ:   mean;   σ:   standard deviation

To compute probabilities:

Enter the values of the mean, standard deviation, and the value of x [Perc (x)] at which the cdf is to be computed; click

Example 1. When mean = 1.0, standard deviation = 2.0, and the observed value [Perc (x)] = 3.5,

To compute percentiles:

Enter the values of the mean, standard deviation, and the cumulative probability [image]; click

Example 2. When mean = 1.0, standard deviation = 2.0, and the cumulative probability = 0.95, the 95th percentile is 4.28971. That is,

To compute moments:

Enter the values of the mean, and standard deviation and click


Chi-square Distribution

n:   degrees of freedom

To compute probabilities:

Enter the value of the degrees of freedom (df), and the value of x [Perc (x)] at which the cdf is to be computed; click

Example 10.6.1   When df = 13.0, and the observed value [Perc (x)]  = 12.3,

To compute percentiles:

Enter the values of the degrees of freedom, and the cumulative probability [image] ; click

Example 10.6.2 When df = 13.0, and the cumulative probability [image] = 0.95,  the 95th percentile is 22.362. That is,

To compute moments:

Enter the value of the df and click

Remark: For the degrees of freedom greater than 100000, a normal approximation is used to compute the cdf as well as the percentiles.


Student's t Distribution
Student’s t Distribution

n:   degrees of freedom

To compute probabilities:

Enter the values of the degrees of freedom (df), and [Perc (x)] at which the cdf is to be computed; click [image].

Example 1    When df = 12.0, and the observed value [Perc (x)]  = 1.3,

To compute percentiles:

Enter the value of the degrees of freedom, and the cumulative probability

 ; click

Example 2    When df = 12.0, and the cumulative probability = P(X <= x) = 0.95,  the 95th percentile is 1.78229. That is,

To compute moments:

Enter the value of the df and click [image].

Remark:    For df > 50000, the following approximation is used to compute the table values.

where Z is the standard normal random variable.


F Distribution

m:    Numerator Degrees of Freedom

n:     Denominator Degrees of Freedom

To compute probabilities:

Enter the values of the numerator degrees of freedom, denominator df,  and the value of x [Perc (x)] at which the cdf is to be computed; click on

Example 1   When numerator df = 3.3, denominator df = 44.5 and the observed value [Perc (x)] = 2.3,

To compute percentiles:

Enter the values of the degrees of freedom, and the cumulative probability [image] ; click

Example 2 When numerator df = 3.3, denominator df = 44.5,and the cumulative probability [image] = 0.95,  the 95th percentile is 2.73281. That is,

To compute moments:

Enter the values of the numerator df, denominator df , and click


Beta Distribution

a:   shape parameter;   b:   shape parameter

To compute probabilities:

Enter the values of the parameters a, and b and the value of x [Perc (x)] at which the cdf is to be computed; click [image] .

Example  1  When a = 2, b = 3, and the observed value

Perc (x)  = 0.4,

To compute percentiles:

Enter the values a, b, and the cumulative probability [image]; click on [image].

Example 2   When a = 2, b = 3, and the cumulative probability P(X <= x) = 0.40, the 40th percentile  is 0.329167. That is,

To compute moments:

Enter the values a and b and click on

Remark 1 When both the shape parameters a and b are greater than 200000,  the usual standard normal approximation is used to compute the table values. That is,


Gamma Distribution

a:  shape parameter;   b:   scale parameter

To compute probabilities:

Enter the values of the shape parameter a, scale parameter

b, and the observed value [Perc (x)]; click on

Example 1 When a = 2, b = 3, and the observed value [Perc (x)] = 5.3,

To compute percentiles:

Enter the values of a, b, and the cumulative probability [image] ; click

Example 2 When a = 2, b = 3, and the cumulative probability [image]  = 0.05, the 5th percentile  is 1.06608. That is,

To compute moments:

Enter the values of a and b; click


Noncentral Chi-square

n = Degrees of Freedom > 0; δ = Noncentrality Parameter > 0

To compute probabilities:

Enter the values of the df, noncentrality parameter, and the observed value x [Perc (x)]; click [image] .

Example 1  When df = 13.0, noncentrality parameter = 2.2 and the observed value [Perc (x)] = 12.3,

To compute percentiles:

Enter the values of the df, noncentrality parameter, and the cumulative probability P(X <= x); click [image].

Example 2 When df = 13.0, noncentrality parameter = 2.2, and the cumulative probability P(X <= x) = 0.95,  the 95th percentile is 26.0113. That is,

To compute moments:

Enter the values of the df, and the noncentrality parameter; click [image].


Noncentral t Distribution

n  = Degrees of Freedom;  δ = Noncentrality Parameter

To compute probabilities:

Enter the values of the degrees of freedom (df), noncentrality parameter, and the value of  x [Perc (x)] at which the cumulative probability is to be computed; click [image] .

Example 1    When df = 13.0, noncentrality parameter = 2.2 and the observed value [Perc (x)] =2.2,

To compute percentiles:

Enter the values of the df, noncentrality parameter, and the cumulative probability P(X <= x); click [image].

Example 2    When df = 13.0, noncentrality parameter = 2.2, and the cumulative probability P(X <= x)  = 0.90,  the 90th percentile is 3.87082. That is,

To compute moments:

Enter the values of the DF, and the noncentrality parameter; click [image].


Noncentral F Distribution

m = Numerator df;   n = Denominator df; δ = Noncentrality Parameter

To compute probabilities:

Enter the values of the numerator df, denominator df, noncentrality parameter, and the observed value x [Perc (x)];  click [image] .

Example 1.     When numerator df = 4.0, denominator df = 32.0, noncentrality parameter = 2.2, and the observed value [Perc (x)] = 2,

To compute percentiles:

Enter the values of the df, noncentrality parameter, and the cumulative probability P(X <= x); click [image].

Example 2.     When numerator df = 4.0, denominator df = 32.0, noncentrality parameter = 2.2, and the cumulative probability P(X <= x) = 0.90,  the 90th percentile is 3.22243. That is,

To compute moments:

Enter the values of the numerator df, denominator df and the noncentrality parameter; click [image].


Laplace Distribution

a = Location Parameter;   b = Scale Parameter

To compute probabilities:

Enter the values of the parameters a, and b and the observed value x [Perc (x)]; click [image] .

Example 1    When a = 3, b = 4, and the observed value

Perc (x) = 4.5,

To compute percentiles:

Enter the values of a, b, and the cumulative probability; click [image].

Example 2    When a = 3, b = 4, and the cumulative probability P(X <= x) = 0.95, the 95th percentile  is 12.2103. That is,

To compute moments:

Enter the values of a and b and click [image].


Logistic Distribution

a = Location Parameter; b = Scale Parameter

To compute probabilities:

Enter the values of the parameters a, and b and the observed value x [Perc (x)]; click [image] .

Example 1  When a = 2, b = 3, and the observed value x = 1.3,

To compute percentiles:

Enter the values a, b, and the cumulative probability; click [image].

Example 2  When a = 2, b = 3, and the cumulative probability  = 0.25, the 25th percentile  is –1.29584. That is,

To compute moments:

Enter the values of a and b and click [image].


Weibull Distribution

m = location parameter;  c = shape parameter;  b = scale parameter

To compute probabilities:

Enter the values of the parameters m, c, and b and the observed value x [Perc (x)]; click [image] .

Example 1  When m = 0, c = 2.3, b = 2, and the observed value [Perc(x)] = 3.4,

To compute percentiles:

Enter the values of m, c, b, and the cumulative probability; click [image].

Example 2  When m = 0, c = 2.3, b = 2, and the cumulative probability P(X <= x) = 0.95, the 95th percentile  is 3.22259. That is,

To compute moments:

Enter the values of c and b and click [image]. The moments are computed assuming that m = 0.


Extreme Value Distribution
Extreme Value

a = location parameter;  b = scale parameter

P(X ≤ x) = exp{-exp[-(x – a)/b]},  -∞ < x < ∞,  b > 0.

To compute probabilities:

Enter the values of the parameters a, and b and the observed value [Perc (x)]; click [image] .

Example 1  When a = 1, b = 2, and the observed value Perc (x) = 1.2,

To compute percentiles:

Enter the values a, b, and the cumulative probability P(X <= x); click [image].

Example 2  When a = 1, b = 2, and the cumulative probability = 0.15, the 15th percentile  is -0.280674. That is,

To compute moments:

Enter the values a and b and click [image].


Pareto Distribution

To compute probabilities:

Enter the values of the parameters a, and b and the observed value [Perc (x)]; click [image] .

Example 1  When a = 2, b = 3, and the observed value Perc (x) = 3.4,

To compute percentiles:

Enter the values a, b, and the cumulative probability P(X <= x); click [image].

Example 2  When a = 2, b = 3, and the cumulative probability = 0.15, the 15th percentile  is 2.11133. That is,

To compute moments:

Enter the values a and b and click [image].


Cauchy Distribution

a = location parameter;  b = scale parameter

To compute probabilities:

Enter the values of the parameters a, and b and the observed value x [Perc (x)]; click [image] .

Example 1  When a = 1, b = 2, and the observed value Perc (x) = 1.2,

To compute percentiles:

Enter the values of a, b, and the cumulative probability; click [image].

Example 2  When a = 1, b = 2, and the cumulative probability P(X <= x) = 0.95, the 95th percentile  is 13.6725. That is,


Distribution of Runs

Example 1.    Consider the sequence:

a  a  a  b  b  a  b  b  b a  a  a  a  b  b  a  a  a  a  b

Here m = 12, n = 8, and the observed number of runs r = 8. To get the probabilities, enter 12 for the number of first type symbols, 8 for the number second type symbols, and 8 for the observed number of runs; click on [image] to get

Since the probability of observing 8 or fewer runs is not less than 0.05/2 = 0.025, the null hypothesis that the arrangement is random will be retained at the 5% level.

To find the critical value at the 5% level, enter 0.025 for the tail probability, and click on [image] to get

This means that the null hypothesis of randomness will be rejected (at 5% level) when the observed number of runs is less than or equal to 6 or greater than or equal to 16. Because of the discreteness of the distribution, the attained size is 0.024609 + 0.00655 =0.031159 (not 0.05).

Example 2.    Suppose that m = 50 and n = 40. To find the 10% critical points, enter 50 for the first type symbols, 40 for the second type symbols, and 0.05 for the tail probability; click on [image] to get 37 and 54. This means that

Thus, the null hypothesis of randomness will be rejected at 10% level, if the total number of runs is less than or equal to 37 or is greater than or equal to 54.


Sign Test and CI for Median

Let  X1, … , Xn  be a sample of independent observations from a continuous population. Let X(j) denote the jth smallest of the Xi’s. For given confidence level (1 - α), and the sample size n, StatCalc computes the integers p and q such that the interval (X(p), X(q))

would contain the population median with 100(1 - α)% confidence. StatCalc also computes the p-values for hypothesis testing about the median.

Example 1    To compute a 95% confidence interval for the median of a continuous population based on a sample of 40 observations, select [image]  from StatCalc, enter 40 for n, 0.95 for confidence level, click [image]  to get 14 and 27. That is, the required confidence interval is formed by the 14th and 27th order statistics from the sample.

Example 2    Suppose that a sample of 40 observations yielded k = 13. To obtain the p-value of the test when Ha: M < M0, select [image]  from StatCalc, enter 40 for n, 13 for k and click [image] to get

Since this p-value is less than 0.05, the null hypothesis that H0: M = M0 will be rejected at the 5% level.


Wilcoxon Signed-Rank Test

Example 1    Suppose that a sample of 20 observations yielded T+ = 130. Then, enter 20 for n, 130 for T+, and click on [image] to get

This program also computes the critical point for given n and the level α.

Example 2    When n = 20, cumulative probability = 0.95, the upper tail critical point is 150; that is

.

Remark :    For n > 60, the standard normal approximation is used to compute the table values.


Wilcoxon Rank-Sum Test

WRS statistic is useful to test whether two continuous distributions are equal. Let X1, … , Xm be independent observations from a continuous distribution FX  and Y1, … , Yn be independent observations from a continuous distribution FY. Pool  X and Y observations, and arrange them in increasing order. Let W denote the sum of the ranks of the X observations in the pooled sample.

For given m, n, and W this program computes the probabilities.

Example 1    When m = 13, n = 12, and the observed value of W is 180,

This program also computes the critical values for given m, n, and the level of significance.

Example 2    When m = 13, n = 12, and the cumulative probability is 0.05, the left tail critical value is 138. Because of the discreteness of the distribution, the attained level is 0.048821; that is

Remark 1    An exact method is used to compute the table values when m ≤ 60 and n ≤ 60. For all other values, the standard normal approximation is used.


Nonparametric Tolerance Limits
NP Tolerance Limits

Let X1 ,…,Xn be a sample of independent observations from a continuous population. Let X(k) denote the kth smallest of the Xi’s. For given 0 < p < 1, 0 < g < 1, this program computes the value of n so that the interval

(X(1), X(n))

would contain at least p proportion of  the population with confidence g.

Example 1    When p = 0.90, and g = 0.95, the value of n is 46; that is the interval (X(1), X(46)) would contain at least 90% of the population with confidence 95%. Further, the sample size required for a one-sided tolerance limit is 29; that is 90% of the population data are less than or equal to X(29) - the largest observation in a sample of 29 observations, with confidence 95%. Similarly, 90% of the population data are greater than or equal to X(1) - the smallest observation in a sample of 29 observations, with confidence 95%.


Tolerance Factor for Normal Population
Tolerance Factors for Normal Population

Let [image] and s denote respectively the mean and standard deviation of a random sample of n observations from a normal population. For given n, df, 0 < p < 1, and 0 < g < 1, this program computes the value of k such that the interval

contains at least 100p% of the population with confidence 100g%. We refer this tolerance interval as (p, g) tolerance interval. StatCalc uses an exact method of computing the tolerance factor k.

Remark 1    Here, the df associated with s is n - 1. In some situations, the df associated with the s could be different from n - 1. For example, in one-way ANOVA,  the df  associated with the pooled sample variance is (n -m), where m denotes the number of groups. If one is interested in computing (p, g) tolerance interval

for the first population, then for this case, sample size is n1 and the degrees of freedom associated with the pooled variance is

where ni denotes the size of the sample from the ith group, i = 1,…, m.

This program also computes one-sided tolerance factor c such that at least 100p% of the population data are less than or equal to

with confidence 100g%. The lower tolerance limit is

.

That is, at least 100p% of the population data are greater than or equal to [image].

Example 1    When n = 23, df = 22, p = 0.90, and g = 0.95, the one-sided tolerance factor is 1.86902, and the two-sided tolerance factor is 2.25125.


2 x 2 Table
Fisher’s Exact Test for 2 x 2 Table

For a given 2 x 2 table, StatCalc computes the probability of observing k or more successes (as well as the probability of observing k or less number of successes) in the cell (1,1). If either of this probability is less than α/2, then the null hypothesis of equal proportions will be rejected at the level of significance α.

Further, for given level of significance, power and guess values on p1 and p2, StatCalc also computes an approximate sample size needed for each group to satisfy the requirements.

To find the p-value, enter the cell values and click Prob <= (1,1) cell .

To compute the sample size, enter the level of significance, power, guess values of

p1 and p2, and then click S Size .


Tolerance Factor for Multivariate Normal Population
Multivariate Tolerance Factors

Let [image] and S denote respectively the mean and variance-covariance matrix of a random sample of n observations from an m-variate normal population. For a given n,  m, 0 < p < 1, and 0 < g < 1, this program computes an approximate tolerance factor k such that the region

would contain at least 100p% of the population with confidence 100g%.

Example 1    When n = 35, m = 3, p = 0.90, and g = 0.95, the 90% content – 95% coverage tolerance factor k is 9.83.


Multiple Correlation Coefficient

To compute probabilities:

Enter the values of the sample size n, number of variates m, squared population multiple correlation coefficient ρ, and the observed value of the squared sample multiple correlation coefficient r2; click

Example 1    When n = 40, m = 4, ρ2 = 0.8, r2 = 0.75,

To compute percentiles:

Enter the values of the sample size n, number of variates m,  squared population multiple correlation coefficient ρ2, and the cumulative probability; click

Example 2    When n = 40, m = 4, ρ2 = 0.8, the 90th   percentile is 0.874521.

To compute confidence intervals:

Enter the values of n, m, r2, and the confidence level; Click

Example 3    When n = 40, m = 4, r2 = 0.75, the 95% confidence interval for ρ2 is (0.544189, 0.848982).


Simple Correlation Coefficient
Correlation Coefficient

To compute probabilities:

Enter the sample size n, population correlation ρ, and the observed sample correlation r; click

Example 1    When n = 30, ρ = 0.7, r = 0.75,

To compute percentiles:

Enter the sample size n, population correlation ρ, and the cumulative probability; click

Example 2    When n = 30, ρ = 0.7, and the cumulative probability is 0.95, the 95th percentile is 0.832097. That is,

To compute confidence intervals:

Enter the sample size, sample correlation, and the confidence level; click

Example 3    When n = 30, r = 0.6, a 95% confidence interval for ρ is (0. 30584, 0. 78959).

Remark  1    An exact method is used to compute the cdf and the percentiles when n in [4, 100] and ρ in [-0.83, 0.83]. For the other values, Fisher's z transformation is used to compute cdf and the percentiles. Further, confidence intervals are based on the Fisher's z transformation.


Bivariate Normal Distribution

Let (X, Y) be a bivariate normal random vector with mean = (0, 0), and the correlation coefficient ρ. For given x, y, and ρ,  this program computes the following probabilities:

Example 1    When x = 1.1, y = 0.8, and ρ = 0.6,

If (X, Y) is a normal random vector with mean = (μ1, μ2), and covariance matrix

then to compute the above probabilities at (x, y), enter the standardized values


Max of t Variables
Maximum of | t | Variables

Let [image] be independent normal random variables with mean μ and common standard deviation σ. Let [image] follow a chi-square distribution with df = m. StatCalc computes the distribution function of

The percentiles of X are useful for constructing simultaneous confidence intervals for the treatment effects and orthogonal estimates in the analysis of variance, and to test extreme values.

To compute probabilities:

Enter the values of the number of groups k, df, and the observed x; click

Example 1.  When k  =  4, df  =  45, and x = 2.3,

To compute percentiles:

Enter the values of k, df, and the cumulative probability; click

Example 2. When k = 4, df = 45, and the cumulative probability is 0.95, the 95th percentile is 2.5897. That is,


Lognormal Distribution

To compute probabilities:

Enter the values of the parameters μ and σ, and the

observed value x [Perc (x)]; click [image].

Example 1  When μ = 1, σ = 2, and the observed value

[Perc (x)] = 2.3,

To compute percentiles:

Enter the values μ, σ, and the cumulative probability P(X <= x); click on [image].

Example 2  When μ = 1, σ = 2, and the cumulative probability P(X <= x) = 0.95, the 95th percentile  is 72.9451. That is,

To compute moments:

Enter the values of μ and σ and click [image].