Exam 14: Nonparametric Statistics Available Online

At six different randomly chosen times, a researcher measures the temperature (in degrees Fahrenheit)of a pint of milk from a supermarketʹs shelf. The measurements are shown below. 36 38 39 39 41 43 a. Suppose the researcher is interested in determining whether the median temperature exceeds $40 ^ { \circ } \mathrm { F }$ . Set up the null and alternative hypotheses of interest. b. How many of the measurements exceed $40 ^ { \circ } \mathrm { F }$ ? c. Assuming that $p = .5$ , find the binomial probability that at least two measurements exceed $40 ^ { \circ } \mathrm { F }$ . d. What do you conclude about $H _ { 0 }$ and $H _ { \mathrm { a } }$ ?

What is referred to as nonparametrics?

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: 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?

Consider the data set shown below. Find the Spearmanʹs Rank Correlation for the variables x and y. 0 4 2 3 8 10 11 -2 0 2 4 6 8 10

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

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: What alternative hypothesis should the instructor test to show that the students in the traditional Class outperformed the students in the online class?

The Wilcoxon rank sum test is used to test the hypothesis that the probability distributions associated with two populations are equivalent.

Specify the rejection region for the Wilcoxon signed rank test in the following situation. $n = 25 , \alpha = .05$ $H _ { 0 }$ : Two probability distributions, 1 and 2, are identical $H _ { \mathrm { a } }$ : Probability distribution of population 1 is shifted to the right of the probability distribution for population 2

Verbal SAT scores for students randomly selected from two different schools are listed below. Use the Wilcoxon rank sum procedure to test the claim that there is no difference in the scores from each school. Use $\alpha = .05$ .

A weight-lifting coach claims that a weight-lifter can increase strength by taking vitamin E. To test the theory, the coach randomly selects 9 athletes and gives them a strength test using a bench press. The results are listed below. Thirty days later, after regular training supplemented by vitamin E, they are tested again. The new results are listed below. Use the Wilcoxon signed rank test to find the critical value to test the claim that the vitamin $\mathrm { E }$ supplement is effective in increasing the athletes' strength. Use $\alpha = .05$ . Athlete 1 2 3 4 5 6 7 8 9 before 204 274 264 206 270 219 226 196 267 After 214 279 264 204 277 234 231 191 272

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?

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

For a valid Spearmanʹs test, the probability distribution for just one of the two variables must be continuous.

A real estate agent surmises that the median rent for a one-bedroom apartment in a beach community is at least $\$ 2000$ per month. The rents for a random sample of 15 one-bedroom apartments are listed below. Test the agent's claim. Use $\alpha = .01$ . \ 2300 \ 2250 \ 1700 \ 1875 \ 1735 \ 2750 \ 2175 \ 1670 \ 2390 \ 3000 \ 1995 \ 2000 \ 2075 \ 2000 \ 1780

Test the hypothesis that the median age of statistics teachers is 46 years. A random sample of 60 statistics teachers found 25 above 46 years and 35 below 46 years. $\text { Use } \alpha = .01 \text {. }$

Fifteen SmartCars were randomly selected and the highway mileage of each was noted. It was desired to determine if the median miles per gallon of all SmartCars exceeded 35 miles per gallon. What type of nonparametric analysis should be used?

The median household income of a community is reported to be $62,000. A random sample of six households in the community yielded the following incomes. $\$ 42,000 \quad \$ 57,000 \quad \$ 65,000 \quad \$ 69,000 \quad \$ 75,000 \quad \$ 150,000$ Does the sample provide sufficient evidence to refute the reported median household income? Perform a sign test using $\alpha = .10$ .

A convenience store owner believes that the median number of lottery tickets sold per day is 47. A random sample of 20 days yields the data below. Test the ownerʹs claim. Use α = .05. 30 46 57 62 29 53 68 25 31 36 45 52 52 42 42 47 47 57 52 36

A physician claims that a person's diastolic blood pressure can be lowered, if, instead of taking a drug, the person listens to a relaxation tape each evening. Ten subjects are randomly selected. Their blood pressures, measured in millimeters of mercury, are listed below. The 10 patients are given the tapes and told to listen to them each evening for one month. At the end of the month, their blood pressures are taken again. The data are listed below. Use the Wilcoxon signed rank test to test the physician's claim. Use $\alpha = .05$ . Patient 1 2 3 4 5 6 7 8 9 10 Before 86 91 87 82 99 90 96 89 92 84 After 83 85 87 74 93 79 99 79 88 68

The results of independent random samples from two populations are shown below. Sample A: 35, 38, 42, 43, 45, 47, 49, 58 Sample B: 36, 41, 44, 53, 57 Calculate the rank sum for each sample. Which would be used as the test statistic in a Wilcoxon rank sum test?