Ch. 9 - Inferences from Two Samples

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446당신이 사용하는 게 아니라요?교과서 변경

Triola 14th Edition

Ch. 9 - Inferences from Two Samples

문제 9.QQ.2

9장, 문제 9.QQ.2

Test Values p_cap1, p_cap2. Find the values of and the pooled proportion p_bar obtained when testing the claim given in Exercise 1.

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Identify the sample proportions \( \hat{p}_1 \) and \( \hat{p}_2 \) from the problem statement or data provided. These represent the proportions of successes in the two samples.

Determine the sample sizes \( n_1 \) and \( n_2 \) for the two groups. These are the total number of observations in each sample.

Calculate the pooled proportion \( \bar{p} \) using the formula: \( \bar{p} = \frac{x_1 + x_2}{n_1 + n_2} \), where \( x_1 = \hat{p}_1 \cdot n_1 \) and \( x_2 = \hat{p}_2 \cdot n_2 \).

Substitute the values of \( x_1 \), \( x_2 \), \( n_1 \), and \( n_2 \) into the formula to compute \( \bar{p} \).

Use the pooled proportion \( \bar{p} \) in further hypothesis testing or calculations as required by the problem.

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주요 개념

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Pooled Proportion

The pooled proportion, denoted as p_bar, is a weighted average of two sample proportions used in hypothesis testing. It combines the successes and failures from both samples to provide a single estimate of the proportion under the null hypothesis. This is particularly useful when comparing two proportions to determine if there is a significant difference between them.

Sample Proportion

The sample proportion, represented as p_cap, is the ratio of the number of successes to the total number of observations in a sample. It serves as an estimate of the true population proportion and is calculated by dividing the count of successes by the sample size. Understanding sample proportions is essential for conducting tests of significance and making inferences about population parameters.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample data to determine whether to reject H0. This process often includes calculating test statistics and p-values to assess the strength of evidence against the null hypothesis.

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Smoking Cessation Programs

a. Construct the confidence interval that could be used to test the claim in Exercise 5. What feature of the confidence interval leads to the same conclusion from Exercise 5?

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Body Temperatures Listed below are body temperatures from six different subjects measured at two different times in a day (from Data Set 5 “Body Temperatures” in Appendix B).

b. Identify the null and alternative hypotheses for using the sample data to test the claim that the differences between 8 AM temperatures and 12 AM temperatures are from a population with a mean equal to 0°F

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Identifying Hypotheses In a randomized clinical trial of adults with an acute sore throat, 288 were treated with the drug dexamethasone and 102 of them experienced complete resolution; 277 were treated with a placebo and 75 of them experienced complete resolution (based on data from “Effect of Oral Dexamethasone Without Immediate Antibiotics vs Placebo on Acute Sore Throat in Adults,” by Hayward et al., Journal of the American Medical Association). Identify the null and alternative hypotheses corresponding to the claim that patients treated with dexamethasone and patients given a placebo have the same rate of complete resolution.

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Variation Find the value of the test statistic used for testing the claim that the two samples from Exercise 5 are from populations having the same variation.

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In Exercises 1–10, based on the nature of the given data, do the following:

a. Pose a key question that is relevant to the given data.

b. Identify a procedure or tool from this chapter or the preceding chapters to address the key question from part (a).

c. Analyze the data and state a conclusion.

IQ Scores of Twins Listed below are IQ scores of twins listed in Data Set 12 “IQ and Brain Size” in Appendix B. The data are pairs of IQ scores from ten different families.

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Smoking Cessation Programs Among 198 smokers who underwent a “sustained care” program, 51 were no longer smoking after six months. Among 199 smokers who underwent a “standard care” program, 30 were no longer smoking after six months (based on data from “Sustained Care Intervention and Postdischarge Smoking Cessation Among Hospitalized Adults,” by Rigotti et al., Journal of the American Medical Association, Vol. 312, No. 7). We want to use a 0.01 significance level to test the claim that the rate of success for smoking cessation is greater with the sustained care program. Test the claim using a hypothesis test.

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