Execution of nonparametric Hypothesis Tests

PART 1

1. a) What is the nonparametric alternative to a 1-sample t test for means?

The non-parametric alternative to a 1-sample t test for means is the Wilcoxon signed-rank test

b) What is the nonparametric alternative to a 2-sample t test for means?

The non-parametric alternative to a 2-sample t test for means is the Wilcoxon 2-sample rank-sum test

c) A test to see if three or more means are equal is called ANOVA for analysis of variance. What is the nonparametric alternative to an ANOVA test for means?

The nonparametric alternative to an ANOVA test for means is the Kruskal-Wallis test

2. Explain the advantages and disadvantages of nonparametric tests

Advantages

i. The probability statements that are attained from majority of non-parametric tests are exact and precise probabilities

ii. If the sample sizes being used in the experiment or study are as minimal as 6, there is no other option but to make use of non-parametric tests with the exception that the nature of the population distribution is correctly known.

iii. There are ideal and appropriate tests for the treatment of the observations from samples obtained from numerous dissimilar populations

iv. Tests are accessible for the treatment of data which are intrinsically in ranks in addition to data whose apparently mathematical scores have solely the strong suit of ranks

v. There are methods accessible for treatment of data which are basic classificatory

vi. The non-parametric tests are considerably simpler to study, understand, and apply as compared to parametric tests (Vaughan, 2001).

Disadvantages

i. Non-parametric tests have low power in comparison to parametric tests. This implies that they have low probability in attaining a significant difference is there exists one

ii. Non-parametric tests place a huge dependence on statistical tables (Weiers, 2010).

3. What distribution is used for the Wilcoxon rank sum test?

The distribution that is used for the Wilcoxon rank sum test is the normal distribution

4. What are the conditions for the Kruskal-Wallis test? 

The conditions for the Kruskal-Wallis test include the following:

i. The assumption made is that the samples drawn from the population are done in a random manner

ii. There is also the assumption that the observations made from the sample are independent of one another

iii. The measurement scale for the dependent variable ought to be at least ordinal

PART 2

1. A marketing firm tests three different brands of energy drinks to see how many equivalents of potassium per quart each contains. The results are given below.

                                  Brand A           Brand B               Brand C

a) Find the median for each brand.

The median value is the middle value of a set of data when ranked in an ascending order

Brand A: 3.2, 4.7, 5.0, 5.1, 5.2

Therefore the median is 5.0

Brand B: 5.3, 6.4, 6.8, 7.2, 7.3

Therefore, the median is 6.8

Brand C: 6.2, 6.3, 6.6, 7.1, 8.2

Therefore, the median is 6.6

b) State the null hypothesis used to test the medians

H0: There is no difference between the medians in the three different brands of energy drinks

c) Perform a Kruskal-Wallis test. Give the value of the test statistic. Then, copy and paste the results of the test into your Word document.

H = 12 / n (n + 1) ? Ri2 / ni – 3 (n + 1)

Sample 1            Sample 2                Sample 3

                                    4.7 - 2                  5.3 - 6                      6.3 - 8

                                    3.2 - 1              6.4 - 9                    8.2 - 15

                                    5.1 - 4                   7.3 - 14                      6.2 - 7

                                    5.2 - 5                   6.8 - 11                       7.1 - 12

                              5.0 - 3                 7.2 - 13                       6.6 - 10

Total: 15 51 52

The samples are ranked as follows:

15 -8.2

H = 12 / n (n + 1) ? Ri2 / ni – 3 (n + 1)

Therefore the test statistic is 7.3

There are 2 degrees of freedom

2. An engineer wishes to determine if the stopping distance for midsize automobiles is different from that of compact automobiles at 75 mph. The data is shown below.

Automobile  1         2          3         4         5         6         7         8         9         10

Midsize     188      190      195       192     186      194      188     187      214        203

Compact    200     211       206      297      198      204     218      212     196        193

a) State the null and alternate hypotheses.

Null Hypothesis: The stopping distance for midsize automobiles is different from that of compact automobiles at 75 mph

Alternate Hypothesis: The stopping distance for midsize automobiles is not different from that of compact automobiles at 75 mph

b) Use a Wilcoxon rank sum test to determine if there is a difference in the stopping distances between Midsize and compact cars. Copy and paste the results of the test into your Word document.

Descriptive Statistics

N

Mean

Std. Deviation

Minimum

Maximum

Midsize

Compact

Ranks

N

Mean Rank

Sum of Ranks

Compact - Midsize

Negative Ranks

2a

Positive Ranks

8b

Ties

0c

Total

a. Compact < Midsize

b. Compact > Midsize

c. Compact = Midsize

Test Statisticsa

Compact - Midsize

Z

-2.041b

Asymp. Sig. (2-tailed)

a. Wilcoxon Signed Ranks Test

b. Based on negative ranks.

c) Include a carefully-worded conclusion in the both statistical terms and in the context of the problem. 

Since the test statistic is less than 0.05 we accept the null hypothsis

3. In a company that was founded on fairness, the human resources manager was asked to perform a study to determine if the females were paid less than the males. If the study indicated that there was a significant difference, then the women would all be given a 5% raise. To perform the study, female and male workers were paired based on years of experience working for the company to determine if there was a difference in the salaries. Their annual salary was then compared. The data is shown below. 

Males         18    43    32    27    15    45    21    22

Females     16    38    35    29    15    46    25    28

a) Use a Wilcoxon signed rank test to determine if there is a difference in the salaries.

Descriptive Statistics

N

Mean

Std. Deviation

Minimum

Maximum

Males

Females

Ranks

N

Mean Rank

Sum of Ranks

Females - Males

Negative Ranks

2a

Positive Ranks

5b

Ties

1c

Total

a. Females < Males

b. Females > Males

c. Females = Males

Test Statisticsa

Females - Males

Z

-.931b

Asymp. Sig. (2-tailed)

a. Wilcoxon Signed Ranks Test

b. Based on negative ranks.

b) Based on the results, address the decision of whether the females should be given a raise. 

Since the test statistic is greater than 0.05 and there is a significant difference, women should be given a raise of 5%