Exam 14: Nonparametric Statistics

Online classes are becoming more and more prevalent at the college level. A statistics instructor randomly sampled ten students from his traditional face-to-face class and ten students from his online class to compare their comprehension of the material that was taught in the class. He administered the same final exam to each student and wants to use the Wilcoxon Rank Sum test to compare their exam scores. The results are shown below: Traditional Class 82917568 93857470 5682 Online Class 67667273 77764881 8692 Calculate the test statistic for the Wilcoxon Rank Sum Test. A) $T _ { 1 } = 44$ B) $T _ { 1 } = 76$ C) $T _ { 1 } = 120$ D) $T _ { 1 } = 60$

A convenience store owner believes that the median number of lottery tickets sold per day is 79. A random sample of 20 days yields the data below. Test the owner's claim. Use α = .05. 62 78 89 94 61 85 100 57 63 68 77 84 84 74 74 79 79 89 84 68

What are distribution-free tests?

A marketing study was conducted to compare the mean age of male and female purchasers of a certain product. Random and independent samples were selected for both male and female purchasers of the product. It was desired to test to determine if the distribution of ages of all female purchasers was shifted to the right of the distribution of ages of all male purchasers. What type of nonparametric analysis should be used?

Eight SmartCars were randomly selected and the highway mileage of each was noted. The highway mileage per gallon for each car is shown below: 32 42 29 34 41 37 38 39 40 It was desired to determine if the median miles per gallon of all SmartCars exceeded 35 miles per gallon. Identify the test statistic that should be used when conducting the Sign Test. A) $S = 38$ B) $S = 6$ C) $S = 2$ D) $5 = 8$

Test the hypothesis that the median age of statistics teachers is 51 years. A random sample of 60 statistics teachers found 25 above 51 years and 35 below 51 years. Use α = .01.

A certain manufacturer is interested in evaluating two alternative manufacturing plans consisting of different machine layouts. Because of union rules, hours of operation vary greatly for this particular manufacturer from one day to the next. Twenty-eight random working days were selected and each plan was monitored and the number of items produced each day was recorded. Some of the collected data is shown below: DAY PLAN 1 OUTPUT PLAN 2 OUTPUT 1 1234 units 1311 units 2 1355 units 1366 units 3 1300 units 1289 units \downarrow \downarrow \downarrow It was desired to test to determine if the distribution of output for plan 2 was shifted to the right of the distribution of output for plan 1 . What type of nonparametric analysis should be used? A) The Sign Test B) The Wilcoxon Rank Sum Test C) The Wilcoxon Signed Rank Test D) The Kruskal-Wallis $\mathrm { H }$ -Test E) The Friedman $\mathrm { F } _ { \mathrm { r } }$ -Test

The time (in minutes) it takes to assemble a computer component for three different machines is listed below. Workers are randomly selected. Use the Kruskal-Wallis test to test the claim that there is no difference in the distribution of the populations. Use α = .05. Machine 1 Machine 2 Machine 3 36 44 32 35 33 29 36 42 33 34 37 35 37 39 34 35 36 31 36 40 41

The advantage of the Friedman $F _ { r } \text {-test }$ is that the measurements within each block do not need to be ranked.

The number of absences and the final grades of 9 randomly selected students from a statistics class are given below. Can you conclude that there is a correlation between the final grade and the number of absences? Use α = .01. Number of Absences 0 3 6 4 9 2 15 8 5 Final Grade 98 86 80 82 71 92 55 76 82

Online classes are becoming more and more prevalent at the college level. A statistics instructor randomly sampled ten students from his traditional face-to-face class and ten students from his online class to compare their comprehension of the material that was taught in the class. He administered the same final exam to each student and wants to use the Wilcoxon Rank Sum test to compare their exam scores. The results are shown below: Traditional Class 82917568 93857470 5682 Online Class 67667273 77764881 8692 Calculate the test statistic for the Wilcoxon Rank Sum Test for Large Samples. A) $z = 0.756$ B) $z = - 0.756$ C) $z = 115$ D) $z = 95$

Nine students took the SAT test. Their scores are listed below. Later, they took a test preparation course and retook the SAT. Their new scores are listed below. Use the Wilcoxon signed rank test to test the claim that the test preparation had no effect on their scores. Use α = .05. Student 1 2 3 4 5 6 7 8 9 Before Score 1000 850 890 830 900 810 820 1080 980 After Score 1020 850 880 870 930 820 810 1120 1000

The Kruskal-Wallis test is valid only when each sample contains five or fewer measurements.

The only assumption necessary to ensure the validity of the sign test is that the probability distribution of measurements is continuous.

The sign test provides inferences about the population median rather than the population mean.

A scientist is hoping to compare the levels of DDT toxin found in three species of fish in a local river. He randomly samples 50 of each species to use in the analysis. For each fish, he measures the amount of DDT toxin present. Ideally he will be able to determine if a difference exists in the population of DDT toxins found in the three fish species. What type of nonparametric analysis should be used?

Calculate or use a table to find the binomial probability $P ( x \geq 20 ) \text { when } n = 25 \text { and } p = .5$ . Also use the normal approximation to calculate the probability.

Independent random samples from two populations are shown in the table. Sample 1 11149 3125 16 Sample 2 191710 22188 1425 Use the Wilcoxon rank sum test to determine whether the data provide sufficient evidence to indicate a shift in the locations of the probability distributions of the sampled populations. Use α = .05.

A scientist is hoping to compare the levels of DDT toxin found in three species of fish in a local river. He randomly samples 7 of each species to use in the analysis. For each fish, he measures the amount of DDT toxin present. The data is shown below: Species 1 2 3 8.4 4.5 0.58 15 4.2 2 25 3 2.2 5.6 2.3 7.4 4.6 2.5 0.35 8.2 6.8 1.9 6.1 5.1 2.7 Ideally he will be able to determine if a difference exists in the population of DDT toxins found in the three fish species. Identify the rejection region that should be used to conduct the Kruskal-Wallis $\mathrm { H } -$ Test at $\alpha = 0.05$ . A) Reject $\mathrm { H } _ { 0 }$ if $\mathrm { H } > 7.81473$ B) Reject $\mathrm { H } _ { 0 }$ if $\mathrm { H } > 9.48773$ C) Reject $\mathrm { H } _ { 0 }$ if $\mathrm { H } > 7.37776$ D) Reject $\mathrm { H } _ { 0 }$ if $\mathrm { H } > 5.99147$