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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 10a
9장, 문제 10a

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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1
Step 1: Define the null and alternative hypotheses. The null hypothesis (H₀) states that the proportion of women spending money is the same for both groups: p₁ = p₂. The alternative hypothesis (H₁) states that the proportion of women spending money is smaller for the group given the single large bill: p₁ < p₂.
Step 2: Identify the sample proportions and sample sizes. For the group given the single large bill, the sample size is n₁ = 75, and the number of women who spent money is x₁ = 60, so the sample proportion is p̂₁ = x₁ / n₁. For the group given smaller bills, the sample size is n₂ = 75, and the number of women who spent money is x₂ = 68, so the sample proportion is p̂₂ = x₂ / n₂.
Step 3: Calculate the pooled proportion. The pooled proportion (p̂) is calculated as: p̂ = (x₁ + x₂) / (n₁ + n₂). This is used because the null hypothesis assumes the proportions are equal.
Step 4: Compute the test statistic. The test statistic for comparing two proportions is given by: z = (p̂₁ - p̂₂) / sqrt(p̂ * (1 - p̂) * (1/n₁ + 1/n₂)). Substitute the values of p̂₁, p̂₂, p̂, n₁, and n₂ into this formula to calculate the z-score.
Step 5: Determine the critical value and make a decision. Using a significance level of 0.05 and a one-tailed test (since the alternative hypothesis is directional), find the critical z-value from the standard normal distribution table. Compare the calculated z-score to the critical value. If the z-score is less than the critical value, reject the null hypothesis; otherwise, fail to reject the null hypothesis.

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주요 개념

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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 no difference, and an alternative hypothesis (H1) that indicates the presence of an effect or difference. The goal is to determine whether the sample data provide sufficient evidence to reject the null hypothesis at a specified significance level, often denoted as alpha (α).
추천 영상:
가이드 코스
06:21
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
04:46
Step 4: State Conclusion Example 4

Proportions and Comparison

In this context, proportions refer to the fraction of women who spent some or all of the money from the different bill formats. Comparing proportions involves analyzing the differences between two groups to determine if one group has a significantly higher or lower proportion than the other. This comparison is often conducted using statistical tests, such as the chi-square test or z-test for proportions, to assess whether observed differences are likely due to chance.
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가이드 코스
10:28
Difference in Proportions: Hypothesis Tests Example 1
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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

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).

d. If c1 equal to or greater than critical value then conclude that sigma2,1 > sigma2,2 If c1 equal to or greater than critical value then conclude that sigma2,2 > sigma2,1. Otherwise, fail to reject the null hypothesis of sigma2,1 = sigma2,2

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교과서 질문

P-VALUE The test statistic of z = 2.14 is obtained when using the data from Exercise 1 and testing the claim that patients treated with dexamethasone and patients given a placebo have the same rate of complete resolution.


a. Find the P-value for the test.

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교과서 질문

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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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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교과서 질문

F Test Statistic


d. Is the F distribution symmetric, skewed left, or skewed right?

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