Ch. 9 - Inferences from Two Samples

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 9 - Inferences from Two Samples

Problem 9.1.13c

Chapter 9, Problem 9.1.13c

Are Seat Belts Effective? A simple random sample of front-seat occupants involved in car crashes is obtained. Among 2823 occupants not wearing seat belts, 31 were killed. Among 7765 occupants wearing seat belts, 16 were killed (based on data from “Who Wants Airbags?” by Meyer and Finney, Chance, Vol. 18, No. 2). We want to use a 0.05 significance level to test the claim that seat belts are effective in reducing fatalities.

c. What does the result suggest about the effectiveness of seat belts?

Verified step by step guidance

Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis states that the fatality rates for occupants wearing seat belts and not wearing seat belts are the same. Mathematically, H₀: p₁ = p₂, where p₁ is the fatality rate for those not wearing seat belts, and p₂ is the fatality rate for those wearing seat belts. The alternative hypothesis states that the fatality rate for those wearing seat belts is lower. Mathematically, H₁: p₁ > p₂.

Step 2: Calculate the sample proportions for each group. For the group not wearing seat belts, the sample proportion is p̂₁ = x₁ / n₁, where x₁ = 31 (number of fatalities) and n₁ = 2823 (total occupants). For the group wearing seat belts, the sample proportion is p̂₂ = x₂ / n₂, where x₂ = 16 and n₂ = 7765.

Step 3: Compute the pooled proportion (p̂) under the assumption that the null hypothesis is true. The pooled proportion is calculated as p̂ = (x₁ + x₂) / (n₁ + n₂), where x₁ and x₂ are the number of fatalities in each group, and n₁ and n₂ are the total number of occupants in each group.

Step 4: Calculate the test statistic using the formula for a two-proportion z-test: z = (p̂₁ - p̂₂) / sqrt(p̂(1 - p̂)(1/n₁ + 1/n₂)). Substitute the values of p̂₁, p̂₂, p̂, n₁, and n₂ into the formula to compute the z-score.

Step 5: Compare the calculated z-score to the critical value for a one-tailed test at the 0.05 significance level (or use the p-value approach). If the z-score is greater than the critical value (or if the p-value is less than 0.05), reject the null hypothesis. Interpret the result: if the null hypothesis is rejected, it suggests that seat belts are effective in reducing fatalities.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) that represents no effect or no difference, and an alternative hypothesis (H1) that indicates the presence of an effect. In this context, the null hypothesis might state that seat belts do not reduce fatalities, while the alternative suggests they do. The outcome of the test will help determine whether to reject or fail to reject the null hypothesis.

Significance Level

The significance level, often denoted as alpha (α), is the threshold for determining whether a result is statistically significant. In this case, a significance level of 0.05 indicates that there is a 5% risk of concluding that seat belts are effective when they are not. If the p-value obtained from the hypothesis test is less than 0.05, it suggests that the observed data is unlikely under the null hypothesis, leading to its rejection in favor of the alternative hypothesis.

Proportion Comparison

Proportion comparison involves analyzing the proportions of a certain outcome within different groups. In this scenario, we compare the fatality rates of occupants wearing seat belts versus those not wearing them. By calculating the proportions of fatalities in each group, we can assess whether there is a statistically significant difference that supports the claim that seat belts are effective in reducing fatalities during car crashes.

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Difference in Proportions: Hypothesis Tests Example 1

Related Practice

Textbook Question

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. (Note: Answers in Appendix D include technology answers based on Formula 9-1 along with “Table” answers based on Table A-3 with df equal to the smaller of n1-1 and n2-1)

Magnet Treatment of Pain People spend around \$5 billion annually for the purchase of magnets used to treat a wide variety of pains. Researchers conducted a study to determine whether magnets are effective in treating back pain. Pain was measured using the visual analog scale, and the results given below are among the results obtained in the study (based on data from “Bipolar Permanent Magnets for the Treatment of Chronic Lower Back Pain: A Pilot Study,” by Collacott, Zimmerman, White, and Rindone, Journal of the American Medical Association, Vol. 283, No. 10). Higher scores correspond to greater pain levels.

c. Does it appear that magnets are effective in treating back pain? Is it valid to argue that magnets might appear to be effective if the sample sizes are larger?

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Textbook Question

Independent Samples Which of the following involve independent samples?

c. Data Set 1 “Body Data” includes a sample of pulse rates of 147 women and a sample of pulse rates of 153 men.

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Textbook Question

Count Five Test for Comparing Variation in Two Populations Repeat Exercise 16 “Blanking Out on Tests,” but instead of using the F test, use the following procedure for the “count five” test of equal variations (which is not as complicated as it might appear).

c. If the sample sizes are equal (n1 = n2) use a critical value of 5. If n1 is not equals to n2 calculate the critical value shown below.

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Textbook Question

Color and Cognition Researchers from the University of British Columbia conducted a study to investigate the effects of color on cognitive tasks. Words were displayed on a computer screen with background colors of red and blue. Results from scores on a test of word recall are given below. Higher scores correspond to greater word recall.

c. Does the background color appear to have an effect on word recall scores? If so, which color appears to be associated with higher word memory recall scores?

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Textbook Question

In Exercises 5–16, use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal.

The Freshman 15 The “Freshman 15” refers to the belief that college students gain 15 lb (or 6.8 kg) during their freshman year. Listed below are weights (kg) of randomly selected male college freshmen (from Data Set 13 “Freshman 15” in Appendix B). The weights were measured in September and later in April.

c. What do you conclude about the Freshman 15 belief?

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Textbook Question

Clinical Trials of OxyContin OxyContin (oxycodone) is a drug used to treat pain, but it is well known for its addictiveness and danger. In a clinical trial, among subjects treated with OxyContin, 52 developed nausea and 175 did not develop nausea. Among other subjects given placebos, 5 developed nausea and 40 did not develop nausea (based on data from Purdue Pharma L.P.). Use a 0.05 significance level to test for a difference between the rates of nausea for those treated with OxyContin and those given a placebo.

c. Does nausea appear to be an adverse reaction resulting from OxyContin?

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