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Chapter. 15. Nonparametric Statistics. Section. 15.1. An Overview of Nonparametric Statistics. Objective. Understand the difference between parametric statistical procedures and nonparametric statistical procedures.

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### Chapter

15

Nonparametric Statistics

### Section

15.1

An Overview of Nonparametric Statistics

• Understand the difference between parametric statistical procedures and nonparametric statistical procedures

Parametric statistical procedures are inferential procedures conducted under the assumption that the underlying distribution of the data belongs to some parametric family of distributions (such as the normal distribution).

Nonparametric statistical procedures are inferential procedures that do not make any assumptions about the underlying distribution of the data. They do not require that the population belong to any particular parametric family of distributions (such as the normal distribution) and, therefore, are often referred to as distribution-free procedures.

• Most of the tests have very few requirements, so it is unlikely that these tests will be used improperly.

• Most of the tests have very few requirements, so it is unlikely that these tests will be used improperly.

• For some nonparametric procedures, the computations are fairly easy.

• Most of the tests have very few requirements, so it is unlikely that these tests will be used improperly.

• For some nonparametric procedures, the computations are fairly easy.

• The procedures can be used for count data or rank data, so nonparametric methods can be used on data, such as the rankings of a movie as excellent, good, fair, or poor.

• Nonparametric procedures are less efficient than parametric procedures. This means that a larger sample size is required when conducting a nonparametric procedure to have the same probability of a Type I error as the equivalent parametric procedure.

• Nonparametric procedures often discard useful information. For example, the sign test uses only the sign of the data and rank tests merely preserve order-the magnitude of the actual data values is lost. As a result, nonparametric procedures are typically less powerful. Recall that the power of a test refers to the probability of making a Type II error. A Type II error occurs when a researcher does not reject the null hypothesis when the alternative hypothesis is true.

• Because fewer requirements must be satisfied to conduct these tests, researchers sometimes use these procedures when parametric procedures can be used.

The lower the efficiency is, the larger the sample size must be for a nonparametric test to have the probability of a Type I error the same as it would be for its equivalent parametric test.

### Section

15.2

Runs Test for Randomness

• Perform a runs test for randomness

A runs test for randomness is used to test whether data have been obtained or occur randomly. A run is a sequence of similar events, items, or symbols that is followed by an event, item, or symbol that is mutually exclusive from the first event, item, or symbol. The number of events, items, or symbols in a run is called its length.

Runs tests are used to test whether it is reasonable to conclude that data occur randomly, not whether the data are collected randomly. For example, we might wonder whether defective parts come off an assembly line randomly or systematically. If broken parts occur systematically (such as every fourth part), we might be led to believe that we have a broken machine. We don’t collect the data randomly; instead, we select 100 consecutive parts. We want to know whether the defective parts in the 100 selected occur randomly.

• Notation Used in Conducting a Runs Test for Randomness

• Let n represent the sample size of which there are two mutually exclusive types.

• Let n1 represent the number of observations of the first type.

• Let n2 represent the number of observations of the second type.

• Let r represent the number of runs.

Parallel Example 1: Notation in a Runs Test for Randomness

The following data represent the league that won the World Series for the years 1996-2007. Let “AL” represent the American League and “NL” represent the National League.

AL NL AL AL AL NL

AL NL AL AL NL AL

Identify the values of n, n1, n2 and r.

Let n represent the number of World Series in the sample. Let n1 represent the number of World Series won by the American League and n2 the number of World Series won by the National League. Lastly, let r represent the number of runs.

Then, there are n =12 World Series in the sample, n1 = 8 World Series won by the American League, n2 =4 World Series won by the National League and r =9 runs.

Small-Sample Case:

If n1≤20 and n2≤20, the test statistic in the runs test for randomness is r, the number of runs.

Large-Sample Case: n1>20 or n2>20, the test statistic in the runs test for randomness is

Small-Sample Case: To find the critical value at the

 = 0.05 level of significance for a runs test, we use Table X if n1≤20 and n2≤20.

Large-Sample Case: If n1>20 or n2>20, the critical value is found from Table V, the standard normal table.

Parallel Example 2: Obtaining Critical Values from Table X

Find the upper and lower critical values if n1=8 and n2=4.

From Table X, the lower critical value is 3 and the upper critical value is 10.

To test the randomness of data, we can use the following steps, provided that

the sample is a sequence of observations recorded in the order of their occurrence, and

the observations can be categorized into two mutually exclusive categories.

Step 1: Assume the data are random. This forms

the basis of the null and alternative hypotheses, which are structured as follows:

H0: The sequence of data is random

H1: The sequence of data is not random

Step 2: Determine a level of significance, ,

based on the seriousness of making a Type I error. The level of significance is used to determine the critical value.

Note: For the small-sample case, we must use the

level of significance  =0.05.

Step 3: Use the number of runs, r, to compute the test statistic.

Step 4: Compare the critical value to the test

statistic.

Step 5: State the conclusion.

Parallel Example 3: Testing for Randomness (Small-Sample Case)

The following data represent the league that won the World Series for the years 1996-2007. Let “AL” represent the American League and “NL” represent the National League.

AL NL AL AL AL NL

AL NL AL AL NL AL

Test the claim that leagues win the World Series in a non-random way at the  = 0.05 level of significance.

The sample is a sequence of observations (which league won the World Series in a particular year) recorded in the order of occurrence. The observations are in two mutually exclusive categories, American League or National League. The requirements for the test are satisfied.

Step 1: We are testing the hypothesis that the sequence of observations is random. Thus,

H0: The sequence of data is random

H1: The sequence of data is not random

Step 2: The level of significance is  = 0.05. The lower critical value is 3 and the upper critical value is 10 (Parallel Example 2).

Step 3: The test statistic is r = 9 (Parallel Example 1).

Step 4: Since the test statistic is between the lower and upper critical values, we do not reject the null hypothesis.

Step 5: There is insufficient evidence to conclude that the World Series were won by the two leagues in a nonrandom way during the years 1996-2007.

### Section

15.3

Inferences About Measures of Central Tendency

• Conduct a one-sample sign test

A one-sample sign test is a nonparametric test that uses data, converted to plus and minus signs, to test a hypothesis regarding the median of a population. Data values equal to the assumed value of the median are ignored during the test.

The test statistic will depend on the structure of the hypothesis test and on the sample size.

Small-Sample Case: (n ≤ 25)

Large-Sample Case: (n > 25)

The test statistic, z, is

where n is the number of minus and plus signs and k is obtained as described in the small-sample case.

Small-Sample Case: To find the critical value for a one-sample sign test, we use Table XI if n ≤ 25.

Large-Sample Case: If n >25, the critical value is found from Table V, the standard normal table. The critical value is always located in the left tail of the standard normal distribution. For a two-tailed test, the critical value is . For a left-tailed or right-tailed test, the critical value is .

To test hypotheses regarding the median of a

population, we use the following steps, provided

that the sample is a random sample.

Step 1: Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways:

Note:M0 is the assumed value of the median.

Step 2: Count the number of observations below M0, and assign them minus (-) signs. Count the number of observations above M0, and assign them plus (+) signs.

Step 3: Select a level of significance, ,

based on the seriousness of making a Type I error. The level of significance is used to determine the critical value. The critical value for small samples (n ≤ 25) is found from Table XI. The

critical value for large samples (n > 25) is found from Table V.

Step 4: Obtain the test statistic, k.

Note that k is the smaller of the number of minus signs and plus signs in the two-tailed test, that k is the number of plus signs in the left-tailed test, and that k is the number of minus signs in the right tailed test. In addition, n is the total number of plus and minus signs.

Step 5: Compare the critical value to the test

statistic.

Step 6: State the conclusion.

Parallel Example 1: Conducting a One-Sample Sign Test (Small-Sample Case)

According to the United States Bureau of Labor Statistics, in 2000, the median tenure of employees with their current employer is 3.5 years. An economist believes that the median has increased since then. To test this claim, he randomly selects 16 employed individuals, determines their length of employment and obtains the following data.

0.3 0.8 0.7 3.2 10.3 1.4 0.2 0.9

3.6 6.3 11.2 12.8 7.3 13.0 3.8 23.6

Test the claim at the  =0.05 level of significance.

The data were obtained from a random sample so the conditions of the test are met.

Step 1: We want to know if the median tenure of employees with their current employer is greater than 3.5 years. This is a right-tailed test.

H0: M=3.5 versus H1: M > 3.5

Step 2: There are 7 observations less than 3.5 and 9 observations greater than 3.5. Thus, we have 7 minus signs and 9 plus signs with n=16.

Step 3: Because this is a right-tailed test and n ≤ 25, we find the critical value at the  = 0.05 level of significance with n=16 to be 4 (see Table XI).

Step 4: The test statistic is the number of minus signs. Thus, k =7.

Step 5: Since the test statistic is greater than the critical value, 4, we do not reject the null hypothesis.

Step 6: There is insufficient evidence to support the hypothesis that the median tenure of employees with their employer is greater than 3.5 years.

### Section

15.4

Inferences About The Difference Between Two Medians: Dependent Samples

• Test a hypothesis about the difference between the medians of two dependent samples

The Wilcoxon Matched-Pairs Signed-Ranks Test is a nonparametric procedure used to test the equality of two population medians by dependent sampling.

The test statistic will depend on the size of the sample and on the alternative hypothesis. Let n represent the number of nonzero differences.

Small-Sample Case: (n ≤ 30)

Large-Sample Case: Test (n > 30)

The test statistic is given by

where T is the test statistic from the small-sample case.

Small-Sample Case:(n ≤ 30)

Using  as the level of significance, the critical value(s) is (are) obtained from Table XII in Appendix A.

Large-Sample Case: ( Testn > 30)

Using  as the level of significance, the critical value(s) is obtained from Table V in Appendix A. The critical value is always in the left tail of the standard normal distribution.

If a hypothesis is made regarding the medians of two

populations, we can use the following steps to test the

hypothesis, provided that

the samples are dependent random samples and

the distribution of the differences is symmetric.

Although tests for verifying the symmetry of data exist,

we do not present them in this text. All the data given

satisfy the second requirement.

Step 1: TestDetermine the null and alternative hypotheses. The hypotheses can be structured in one of three ways:

Note:MD is the median of the differences of matched pairs.

Step 2: TestCompute the differences in the matched-pairs observations. Rank the absolute value of all sample differences from smallest to largest after discarding those differences that equal 0. Handle ties by finding the mean of the ranks for tied values. Assign negative values to the ranks where the differences are negative and positive values to the ranks where the differences are positive. Find the sum of the positive ranks, T+, and the sum of the negative ranks T-.

Step 3: Test Draw a boxplot of the differences to compare the sample data from the two populations. This helps to visualize the difference in the medians.

Step 4: Test Choose a level of significance, ,

based on the seriousness of making a Type I error. The level of significance is used to determine the critical value. The critical value is found from Table XII for small samples (n ≤ 30). The critical

value is found from Table V for large samples (n > 30).

Step 5: Test Compute the test statistic.

Small-Sample Case (n ≤ 30)

Large-Sample Case ( Testn > 30)

where T is the test statistic from the small-sample case.

Step 6: Test Compare the critical value with the test

statistic.

Step 7: Test State the conclusion.

Parallel Example 1: TestWilcoxon Matched-Pairs Signed-Ranks Test (Small-Sample Case)

The data on the following slide represent the cost of a one night stay in Hampton Inn Hotels and La Quinta Inn Hotels for a random sample of 10 cities. Test the claim that Hampton Inn Hotels are priced differently than La Quinta Hotels at the =0.05 level of significance.

Solution Test

The data were obtained randomly. We assume that the symmetry requirement is satisfied.

Step 1: We want to know if the hotels are priced differently. This is a two-tailed test.

H0: MD=0 versus H1: MD≠ 0

Solution Test

Step 2: In order to calculate T+ and T-, we must find the differences, rank them, and then attach the sign of the difference to the ranks. The differences and their signed ranks are given in the next slide.

Solution Test

Step 2: From the previous slide, we find that T+=54 and |T-| = 1.

Solution Test

Step 3: The figure below shows a boxplot of the differences. The boxplot indicates that the sample-median difference is about 51.

Solution Test

Step 4: We are testing the hypothesis at the =0.05 level of significance. Since this is a two-tailed test and the sample size is less than 30, we find the critical value with n=10 by using Table XII and obtain T0.025=8.

Step 5: The test statistic is the smaller of T+ and |T-| which is 1.

Solution Test

Step 6: The test statistic is less than the critical value (1< 8), so we reject the null hypothesis.

Step 7: There is sufficient evidence at the =0.05 level of significance to conclude that the median room price at Hampton Inns is different than the median room price at La Quinta Inns.

### Section Test

15.5

Inferences About The Difference Between Two Medians: Independent Samples

Objective Test

• Test a hypothesis about the difference between the medians of two independent samples

The TestMann-Whitney Test is a nonparametric procedure that is used to test the equality of two population medians from independent samples.

The test statistic will depend on the size of the samples from each population. Let n1 represent the sample size for population X and n2 represent the sample size for population Y.

Small-Sample Case: (n1≤ 20 and n2≤ 20 )

If S is the sum of the ranks corresponding to the sample from population X, then the test statistic, T, is given by

Note: The value of S is always obtained by summing the ranks of the sample data that correspond to Mx, the median of population X, in the hypothesis.

Large-Sample Case: Test (n1> 20 or n2> 20 )

From the Central Limit Theorem, the test statistic is given by

where T is the test statistic from the small-sample case.

Small-Sample Case:(n1≤ 20 and n2≤ 20)

Using  as the level of significance, the critical value(s) is(are) obtained from Table XIII in Appendix A.

Large-Sample Case: ( Testn1> 20 or n2> 20 )

Using  as the level of significance, the critical value(s) is(are) obtained from Table V in Appendix A.

To test hypotheses regarding the medians of two

populations, we can use the following steps provided that

the samples are independent random samples and

the shape of the distributions are the same.

Throughout this section, we will assume that the

condition that the shape of the distributions be the same

is satisfied.

Step 1: Test Draw a side-by-side boxplot to compare the sample data from the two populations. This helps to visualize the difference in the medians.

Step 2: TestDetermine the null and alternative hypotheses. The hypotheses are structured as follows:

Note:Mx is the median of population X and My is the median of population Y.

Step 3: TestRank all sample observations from smallest to largest. Handle ties by finding the mean of the ranks for tied values. Find the sum of the ranks for the sample from population X.

Step 4: Test Choose a level of significance, ,

to match the seriousness of making a Type I error. The level of significance is used to determine the critical value. The critical value is found from Table XIII for small samples (n1≤ 20 and n2≤ 20) and from Table V for large samples

(n1 > 20 or n2 > 20).

Step 5: Test Compute the test statistic. Note that S is the sum of the ranks obtained from the sample observations from population X. In addition, n1 is the size of the sample from population X, and n2 is the size of the sample from population Y.

Step 6: Test Compare the critical value with the test

statistic.

Step 7: Test State the conclusion.

Parallel Example 1: TestMann-Whitney Test (Small-Sample Case)

A researcher wanted to know whether “state” quarters had a weight that is more than “traditional” quarters. He randomly selected 18 “state” quarters and 16 “traditional” quarters, weighed each of them and obtained the following data.

Parallel Example 1: TestMann-Whitney Test (Small-Sample Case)

Test the claim that state quarters have a higher median weight than traditional quarters at the =0.05 level of significance.

Solution Test

Step 1:

Solution Test

Step 1: Based on the boxplots, the median weight for the state quarters is higher. We want to estimate whether this difference is due to differences in the population medians or to sampling error.

Solution Test

Step 2: We want to know if the median weight for the state quarters is higher than the median weight of the traditional quarters. This is a right-tailed test.

versus

Solution Test

Step 3: In order to calculate the test statistic, we combine the two data sets into one data set and arrange the data in ascending order. Ranks are shown on the following slide.

Solution Test

Step 3: After ranking the observations, we add up the ranks corresponding to the state quarters to obtain

S = 20+27.5+20+9.5+27.5+34+33+20+ 27.5+

13.5+6+13.5+8+9.5+27.5+23+32+24.5

=376.5

Solution Test

Step 4: Since this is a right-tailed test and both sample sizes are less than 20, we determine the right critical value with n1=18 and n2=16 at the =0.05 level of significance from Table XIII and obtain w0.95 = n1n2-w0.05 = 18(16)-96 = 192.

Solution Test

Step 5: The test statistic is

Solution Test

Step 6: Since the test statistic is greater than the critical value (205.5 > 192), we reject the null hypothesis.

Step 7: There is sufficient evidence at the  = 0.05 level of significance to conclude that the median weight of “state” quarters is greater than that of “traditional” quarters.

### Section Test

15.6

Spearman’s Rank-Correlation Test

Objective Test

• Perform Spearman’s rank-correlation test

The TestSpearman’s rank-correlation test is a nonparametric procedure that is used to test hypotheses regarding the association between two variables.

Test Statistic for Spearman’s Rank-Correlation Test

The test statistic will depend on the size of the sample, n, and on the sum of the squared differences

where di= the difference in the ranks of the two observations in the ith ordered pair.

The test statistic, rs, is also called Spearman’s rank-correlation coefficient.

CAUTION! Rank-Correlation Test

means to square the differences first and then add up the squared differences.

Critical Value for Spearman’s Rank-Correlation Test

Using  as the level of significance, the critical value(s) is(are) obtained from Table XIV in Appendix A. For a two-tailed test, be sure to divide the level of significance, , by 2.

Spearman’s Rank-Correlation Test Rank-Correlation Test

To test hypotheses regarding the association between two

variables X and Y, we can use the following steps,

provided that

the data are a random sample of n ordered pairs and

each pair of observations is two measurements taken on the same individual.

Notice that there is no requirement about the form of the

distribution of the data.

Step 1: Rank-Correlation TestDetermine the null and alternative hypotheses which are structured as follows:

Step 2: Rank-Correlation TestRank the X-values and rank the Y-values.

Compute the differences between ranks and then square these differences. Compute the sum of the squared differences.

Step 3: Rank-Correlation Test Choose a level of significance, ,

based on the seriousness of making a Type I error. The level of significance is used to determine the critical value. The critical value is found in Table XIV.

Step 4: Rank-Correlation Test Compute the test statistic.

where n is the sample size and di is the difference in the ranks of the two observations in the ith ordered pair.

Step 5: Rank-Correlation Test Compare the critical value with the test statistic.

Step 6: Rank-Correlation Test State the conclusion.

Parallel Example 1: Rank-Correlation TestSpearman’s Rank-Correlation Test

Is the price of a sport’s car associated with its performance? The following data represent the ranks of the price and performance of 8 sport’s cars. Using Spearman’s Rank Correlation, determine if the two variables are associated at the  = 0.05 level of significance.

Solution Rank-Correlation Test

Step 1: We are looking for evidence that price and performance of sport’s cars are associated. Let X represent the price of the sport’s car and Y represent performance. The null and alternative hypotheses are as follows:

H0: X and Y are not associated

H1: X and Y are associated

Solution Rank-Correlation Test

Step 2: Rank the X-values and rank the Y-values. Compute the differences in ranks and then square the differences. Calculate the sum of the squared differences to obtain . Details are on the following slide.

Solution Rank-Correlation Test

Step 3: This is a two-tailed test with n=8 and  = 0.05. From Table XIV we determine the critical value to be 0.738.

Solution Rank-Correlation Test

Step 4: The test statistic is

Solution Rank-Correlation Test

Step 5: Since the test statistic is less than the critical value and greater than the negative of the critical value (-0.738 < 0.262 < 0.738), we fail to reject the null hypothesis.

Step 6: There is insufficient evidence at the  = 0.05 level of significance to conclude that the price and performance of sport’s cars are associated.

Large-Sample ( Rank-Correlation Testn > 100) Approximation

If n > 100, the test statistic for Spearman’s Rank-Correlation Test is

Compare this test statistic with the critical value obtained from the standard normal table, Table V. For a two-

tailed test, the critical values are . When testing for positive association, the critical value is . When testing for negative association, the critical value is .

### Section Rank-Correlation Test

15.7

Kruskal-Wallis Test

Objective Rank-Correlation Test

• Test a hypothesis using the Kruskal-Wallis test

The Rank-Correlation TestKruskal-Wallis Test is a nonparametric procedure that is used to test whether k independent samples come from populations with the same distribution.

Test Statistic for the Kruskal-Wallis Test Rank-Correlation Test

The test statistic for the Kruskal-Wallis test is

• A computational formula for the test statistic is Rank-Correlation Test

• where

• Ri is the sum of the ranks of the ith sample

• is the sum of the ranks squared for the first sample

• is the sum of the ranks squared for the second sample, and so on

• n1 is the number of observations in the first sample

• n2 is the number of observations in the second sample, and so on

• N is the total number of observations (N=n1+n2+···+nk)

• k is the number of populations being compared

Critical Value for Kruskal-Wallis Test Rank-Correlation Test

Small-Sample Case

When three populations are being compared and when the sample size from each population is 5 or less, the critical value is obtained from Table XV in Appendix A.

Large-Sample Case

When four or more populations are being compared or the sample size from one population is more than 5, the critical value is with k-1 degrees of freedom, where k is the number of populations and  is the level of significance.

• Kruskal-Wallis Test Rank-Correlation Test

• To test hypotheses regarding the distribution of three

• or more populations, we can use the following steps,

• provided that two requirements are satisfied:

• The samples are independent random samples

• The data can be ranked

Step 1: Draw side-by-side boxplots to compare the sample data from the populations. Doing so helps to visualize the differences, if any, between the medians.

Step 2: Rank-Correlation TestState the null and alternative hypotheses, which are structured as follows:

H0: the distributions of the populations are the same

H1: the distributions of the populations are not the same

Step 3: Rank all sample observations from smallest to largest. Handle ties by finding the mean of the ranks for tied values. Find the sum of the ranks for each sample.

Step 4: Rank-Correlation Test Choose a level of significance, ,

to match the seriousness of making a Type I error. The level of significance is used to determine the critical value. The critical value is found from Table XV for small samples. The critical value is

with k-1 degrees of freedom (found in Table VII) for large samples.

Step 5: Rank-Correlation Test Compute the test statistic.

Step 6: Rank-Correlation Test Compare the critical value to the test

statistic. We reject the null hypothesis if the test statistic is greater than the critical value.

Step 7: Rank-Correlation Test State the conclusion.

Parallel Example 1: Rank-Correlation TestKruskal-Wallis Test

The following data represent the weight (in grams) of pennies minted at the Denver mint in 1990, 1995, and 2000. Test the claim that the distribution of penny weights differs for the three years at the  = 0.05 level of significance.

1990 1995 2000 Rank-Correlation Test

2.50 2.52 2.50

2.50 2.54 2.48

2.49 2.50 2.49

2.53 2.48 2.50

2.46 2.52 2.48

2.50 2.50 2.52

2.47 2.49 2.51

2.53 2.53 2.49

2.51 2.48 2.51

2.49 2.55 2.50

2.48 2.49 2.52

Solution Rank-Correlation Test

The samples are independent random samples that can be ranked. Therefore, the conditions for the Kruskal-Wallis test are met.

Solution Rank-Correlation Test

Step 1: Based on boxplots of the data, the medians do not appear to differ significantly.

Solution Rank-Correlation Test

Step 2: We are interested in determining whether the distribution of penny weights differs for the three years.

The null and alternative hypotheses are as follows:

H0: the distribution of penny weights are the same for the three years

H1: the distribution of penny weights are not the same for the three years

Solution Rank-Correlation Test

Step 3: The ranks of the pennies are given in parentheses.

1990 1995 2000

2.50 (17.5) 2.52 (26.5) 2.50 (17.5)

2.50 (17.5) 2.54 (32) 2.48 (5)

2.49 (10.5) 2.50 (17.5) 2.49 (10.5)

2.53 (30) 2.48 (5) 2.50 (17.5)

2.46 (1) 2.52 (26.5) 2.48 (5)

2.50 (17.5) 2.50 (17.5) 2.52 (26.5)

2.47 (2) 2.49 (10.5) 2.51 (23)

2.53 (30) 2.53 (30) 2.49 (10.5)

2.51 (23) 2.48 (5) 2.51 (23)

2.49 (10.5) 2.55 (33) 2.50 (17.5)

2.48 (5) 2.49 (10.5) 2.52 (26.5)

Solution Rank-Correlation Test

Step 3: We sum the ranks for each of the three years to obtain the following:

Solution Rank-Correlation Test

Step 4: Since the sample sizes for each population are greater than 5, we find the critical value from the chi-square distribution with k-1=3-1=2 degrees of freedom with  = 0.05. Thus, the critical value is

Solution Rank-Correlation Test

Step 5: Note that N=11+11+11=33. The test statistic is

Solution Rank-Correlation Test

Step 6: Since the test statistic is less than the critical value, we fail to reject the null hypothesis.

Step 7: There is insufficient evidence at the  = 0.05 level of significance to conclude that the distribution of penny weights differs for the years 1990, 1995 and 2000.