10. Hypothesis Testing for Two Samples

For ${\bar{x}}_1=40,{\sigma }_1=8.3,n_1=40,{\bar{x}}_2=39,{\sigma }_2=5.8$, & $n_2=50$, perform a hypothesis test to test the claim that ${\mu }_1>{\mu }_2$ for $\alpha =0.05$.

In Exercises 11–14, test the claim about the difference between two population means and at the level of significance . Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1=μ2; α=0.1

Population statistics:σ1=3.4 and σ2=1.5

Sample Statistics: x̅1=16, n1=29, x̅2=14, n2=28

In Exercises 11–14, test the claim about the difference between two population means and at the level of significance . Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1>μ2; α=0.10

Population statistics:σ1=40 and σ2=15

Sample Statistics: x̅1=500, n1=100, x̅2=495, n2=75

Testing the Difference Between Two Means In Exercises 15–24, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

Bed-in-a-Box To compare customer satisfaction with mattresses that are delivered compressed in a box and traditional mattresses, a researcher randomly selects 30 ratings of mattresses in boxes and 30 ratings of traditional mattresses. The mean rating of mattresses in boxes is 68.7 out of 100. Assume the population standard deviation is 6.6. The mean rating of traditional mattresses is 70.9 out of 100. Assume the population standard deviation is 5.6. At α=0.01, can the researcher support the claim that the mean rating of traditional mattresses is greater than the mean rating of mattresses in a box? (Adapted from Consumer Reports)

Testing the Difference Between Two Means In Exercises 15–24, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

Repair Costs: Washing Machines You want to buy a washing machine, and a salesperson tells you that the mean repair costs for Model A and Model B are equal. You research the repair costs. The mean repair cost of 24 Model A washing machines is \$208. Assume the population standard deviation is \$18. The mean repair cost of 26 Model B washing machines is \$221. Assume the population standard deviation is \$22. At α=0.01, can you reject the salesperson’s claim?

Testing the Difference Between Two Means In Exercises 15–24, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

Braking Distances To compare the dry braking distances from 60 to 0 miles per hour for two makes of automobiles, a safety engineer conducts braking tests for 16 compact SUVs and 11 midsize SUVs. The mean braking distance for the compact SUVs is 131.8 feet. Assume the population standard deviation is 5.5 feet. The mean braking distance for the midsize SUVs is 132.8 feet. Assume the population standard deviation is 6.7 feet. At α=0.10 , can the engineer support the claim that the mean braking distances are different for the two categories of SUVs? (Adapted from Consumer Reports)

Testing the Difference Between Two Means In Exercises 15–24, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

Home Prices A real estate agency says that the mean home sales price in Casper, Wyoming, is the same as in Cheyenne, Wyoming. The mean home sales price for 35 homes in Casper is \$349,237. Assume the population standard deviation is \$158,005. The mean home sales price for 41 homes in Cheyenne is \$435,244. Assume the population standard deviation is \$154,716. At α=0.01, is there enough evidence to reject the agency’s claim? (Adapted from Realtor.com)

"Testing the Difference Between Two Means In Exercises 15–24, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

[APPLET] Precipitation A climatologist claims that the precipitation in Seattle, Washington, was greater than in Birmingham, Alabama, in a recent year. The daily precipitation amounts (in inches) for 30 days in a recent year in Seattle are shown below. Assume the population standard deviation is 0.25 inch.

0.00 0.00 0.05 0.01 0.21 0.00 0.00 0.52 0.00 0.010.00 0.19 0.00 0.18 0.02 0.02 0.13 0.00 0.03 0.000.04 0.00 0.41 0.23 0.00 0.80 0.15 0.00 0.00 0.79

The daily precipitation amounts (in inches) for 30 days in a recent year in Birmingham are shown below. Assume the population standard deviation is 0.52 inch.

0.00 0.96 0.84 0.00 0.10 0.00 0.00 0.20 0.00 0.54 0.97 0.00 0.35 0.02 0.04 0.70 0.00 0.00 0.00 0.00 0.03 0.01 0.15 0.27 0.00 0.00 0.93 0.00 0.89 0.01

At α=0.05, can you support the climatologist’s claim? (Source: NOAA)"

Constructing Confidence Intervals for μ1-μ2. You can construct a confidence interval for the difference between two population means μ1-μ2 , as shown below, when both population standard deviations are known, and either both populations are normally distributed or both n1>= 30 and n2>=30 . Also, the samples must be randomly selected and independent.

In Exercises 29 and 30, construct the indicated confidence interval for μ1-μ2 .

Architect Salaries Construct a 99% confidence interval for the difference between the mean annual salaries of entry level architects in Denver, Colorado, and Lincoln, Nebraska, using the data from Exercise 28.

Popcorn Researchers Brian Wansink and Junyong Kim randomly gave 157 moviegoers a free medium (120 grams) or large (250 gram) bucket of popcorn before entering a movie. After the show, the researchers measured how much popcorn the moviegoers consumed. The 77 individuals randomly assigned the medium bucket had a mean consumption of 58.9 grams with a standard deviation of 16.7 grams. The 80 individuals randomly assigned the large bucket had a mean consumption of 85.6 grams with a standard deviation of 14.1 grams. With 95% confidence, determine how much more popcorn was consumed by individuals given the large bucket of popcorn. What is the implication? Source: Wansink, B. Junyong, K. “Bad Popcorn in Big Buckets: Portion Size Can Influence Intake as Much as Taste.” Journal of Nutrition Education & Behavior, September 2005; 35(5):242–245.

College Skills The Collegiate Learning Assessment Plus is an exam that is meant to assess the intellectual gains made between one’s freshman and senior year of college. The exam, graded on a scale of 400 to 1600, assesses critical thinking, analytical reasoning, document literacy, writing, and communication. The exam was administered to 135 freshman in Fall 2012 at California State University Long Beach (CSULB). The mean score on the exam was 1191 with a standard deviation of 187. The exam was also administered to graduating seniors of CSULB in Spring 2013. The mean score was 1252 with a standard deviation of 182. Explain the type of analysis that could be applied to these data to assess whether CLA+ scores increase while at CSULB. Explain the shortcomings in the data available and provide a better data collection technique.

A random sample of size n = 13 obtained from a population that is normally distributed results in a sample mean of 45.3 and sample standard deviation of 12.4. An independent sample of size n = 18 obtained from a population that is normally distributed results in a sample mean of 52.1 and sample standard deviation of 14.7. Does this constitute sufficient evidence to conclude that the population means differ at the α = 0.05 level of significance?

Bribe ‘em with Chocolate In a study published in the journal Teaching of Psychology, the article “Fudging the Numbers: Distributing Chocolate Influences Student Evaluations of an Undergraduate Course” states that distributing chocolate to students prior to teacher evaluations increases results. The authors randomly divided three sections of a course taught by the same instructor into two groups. Fifty of the students were given chocolate by an individual not associated with the course and 50 of the students were not given chocolate. The mean score from students who received chocolate was 4.2, while the mean score for the nonchocolate groups was 3.9. Suppose that the sample standard deviation of both the chocolate and nonchocolate groups was 0.8. Does chocolate appear to improve teacher evaluations? Use the α = 0.01 level of significance.

Vitamin A Supplements in Low-Birth-Weight Babies Low-birth-weight babies are at increased risk of respiratory infections in the first few months of life and have low liver stores of vitamin A. In a randomized, double-blind experiment, 130 low-birth-weight babies were randomly divided into two groups. Subjects in group 1 (the treatment group, n1=65) were given 25,000 IU of vitamin A on study days 1, 4, and 8 where study day 1 was between 36 and 60 hours after delivery. Subjects in group 2 (the control group, n2=65) were given a placebo. The treatment group had a mean serum retinol concentration of 45.77 micrograms per deciliter (μg/dL), with a standard deviation of 17.07 μg/dL. The control group had a mean serum retinol concentration of 12.88 μg/dL, with a standard deviation of 6.48 μg/dL. Does the treatment group have a higher standard deviation for serum retinol concentration than the control group at the α=0.01 level of significance? It is known that serum retinol concentration is normally distributed. 

In a study conducted to determine the role that sleep disorders play in academic performance, researcher Jane Gaultney conducted a survey of 1845 college students to determine if they had a sleep disorder (such as narcolepsy, insomnia, or restless leg syndrome). Of the 503 students with a sleep disorder, the mean grade point average was 2.65 with a standard deviation of 0.87. Of the 1342 students without a sleep disorder, the mean grade point average was 2.82 with a standard deviation of 0.83. Source: SLEEP 2010: Associated Professional Sleep Societies 24th Annual Meeting. 

b. Is there evidence to suggest sleep disorders adversely affect one’s GPA at the α=0.05 level of significance?