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Ch. 9 - Inferences from Two Samples
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Triola 14th EditionCh. 9 - Inferences from Two Samples문제 12a
9장, 문제 12a

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.


a. Use a hypothesis test.

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Step 1: Define the null and alternative hypotheses. The null hypothesis (H₀) states that there is no difference in the rates of nausea between the OxyContin group and the placebo group. Mathematically, H₀: p₁ = p₂, where p₁ is the proportion of nausea in the OxyContin group and p₂ is the proportion of nausea in the placebo group. The alternative hypothesis (Hₐ) states that there is a difference in the rates of nausea, i.e., Hₐ: p₁ ≠ p₂.
Step 2: Calculate the sample proportions for each group. For the OxyContin group, the proportion is p̂₁ = x₁ / n₁, where x₁ = 52 (number of subjects with nausea) and n₁ = 52 + 175 (total subjects in the OxyContin group). For the placebo group, the proportion is p̂₂ = x₂ / n₂, where x₂ = 5 and n₂ = 5 + 40.
Step 3: Compute the pooled proportion (p̂) under the null hypothesis. The pooled proportion is calculated as p̂ = (x₁ + x₂) / (n₁ + n₂), where x₁ and x₂ are the number of subjects with nausea in each group, and n₁ and n₂ are the total number of subjects in each group.
Step 4: Calculate the test statistic using the formula for the z-test for two proportions: z = (p̂₁ - p̂₂) / √[p̂(1 - p̂)(1/n₁ + 1/n₂)]. Substitute the values of p̂₁, p̂₂, p̂, n₁, and n₂ into the formula to compute the z-value.
Step 5: Compare the calculated z-value to the critical z-value for a two-tailed test at the 0.05 significance level (±1.96). If the calculated z-value falls outside the range of -1.96 to 1.96, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Additionally, you can calculate the p-value and compare it to 0.05 to make the decision.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. It involves formulating a null hypothesis (H0) that represents no effect or difference, and an alternative hypothesis (H1) that indicates the presence of an effect or difference. The goal is to determine whether there is enough evidence to reject the null hypothesis at a specified significance level, often denoted as alpha (α).
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Step 1: Write Hypotheses

Significance Level

The significance level, commonly set at 0.05, is the threshold for determining whether the results of a hypothesis test are statistically significant. It represents the probability of rejecting the null hypothesis when it is actually true (Type I error). A significance level of 0.05 indicates that there is a 5% risk of concluding that a difference exists when there is none, guiding researchers in making decisions based on their data.
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가이드 코스
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Step 4: State Conclusion Example 4

Contingency Table

A contingency table is a matrix used to display the frequency distribution of variables, particularly in categorical data analysis. In this context, it helps compare the occurrence of nausea between two groups: those treated with OxyContin and those given a placebo. By organizing the data into a contingency table, researchers can easily calculate proportions and perform statistical tests, such as the chi-square test, to assess the relationship between the treatment and the outcome.
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가이드 코스
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Finding Standard Normal Probabilities using z-Table
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교과서 질문

Denomination Effect A trial was conducted with 75 women in China given a 100-yuan bill, while another 75 women in China were given 100 yuan in the form of smaller bills (a 50-yuan bill plus two 20-yuan bills plus two 5-yuan bills). Among those given the single bill, 60 spent some or all of the money. Among those given the smaller bills, 68 spent some or all of the money (based on data from “The Denomination Effect,” by Raghubir and Srivastava, Journal of Consumer Research, Vol. 36). We want to use a 0.05 significance level to test the claim that when given a single large bill, a smaller proportion of women in China spend some or all of the money when compared to the proportion of women in China given the same amount in smaller bills.


c. If the significance level is changed to 0.01, does the conclusion change?

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교과서 질문

In Exercises 17–24, use the indicated Data Sets from Appendix B. The complete data sets can be found at www.TriolaStats.com. Assume that the paired sample data are simple random samples and the differences have a distribution that is approximately normal.


Heights of Presidents Repeat Exercise 12 “Heights of Presidents” using all of the sample data from Data Set 22 “Presidents” in Appendix B.

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교과서 질문

Denomination Effect A trial was conducted with 75 women in China given a 100-yuan bill, while another 75 women in China were given 100 yuan in the form of smaller bills (a 50-yuan bill plus two 20-yuan bills plus two 5-yuan bills). Among those given the single bill, 60 spent some or all of the money. Among those given the smaller bills, 68 spent some or all of the money (based on data from “The Denomination Effect,” by Raghubir and Srivastava, Journal of Consumer Research, Vol. 36). We want to use a 0.05 significance level to test the claim that when given a single large bill, a smaller proportion of women in China spend some or all of the money when compared to the proportion of women in China given the same amount in smaller bills.


a. Test the claim using a hypothesis test.

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교과서 질문

Denomination Effect A trial was conducted with 75 women in China given a 100-yuan bill, while another 75 women in China were given 100 yuan in the form of smaller bills (a 50-yuan bill plus two 20-yuan bills plus two 5-yuan bills). Among those given the single bill, 60 spent some or all of the money. Among those given the smaller bills, 68 spent some or all of the money (based on data from “The Denomination Effect,” by Raghubir and Srivastava, Journal of Consumer Research, Vol. 36). We want to use a 0.05 significance level to test the claim that when given a single large bill, a smaller proportion of women in China spend some or all of the money when compared to the proportion of women in China given the same amount in smaller bills.


b. Test the claim by constructing an appropriate confidence interval.

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