91Ó°ÊÓ

The Interactive Advertising Bureau surveyed a representative sample of 1000 adult Americans and a representative sample of 1000 adults in China (“Majority of Digital Users in U.S. and China Regularly Shop and Purchase via E-Commerce," November \(10,2016,\) www.iab.com, retrieved December 15,2016 ). They reported that American shoppers are much more likely to use a credit or a debit card to make an online purchase. This conclusion was based on finding that \(63 \%\) of the people in the United States sample said they pay with a credit or a debit card, while only \(34 \%\) of those in the China sample said that they used a credit card or a debit card to pay for online purchases. To determine if the stated conclusion is justified, you want to carry out a test of hypotheses to determine if there is convincing evidence that the proportion who pay with a credit card or a debit card is greater for adult Americans than it is for adult Chinese. a. What hypotheses should be tested to answer the question of interest? b. Are the two samples large enough for the large-sample test for a difference in population proportions to be appropriate? Explain. c. Based on the following Minitab output, what is the value of the test statistic and what is the value of the associated \(P\) -value? If a significance level of 0.01 is selected for the test, will you reject or fail to reject the null hypothesis? d. Interpret the result of the hypothesis test in the context of this problem.

Step by step solution

01

Identify the hypotheses to be tested

To answer the question of interest, we will test the following hypotheses: - Null hypothesis (H0): The proportion of adult Americans who pay with a credit or debit card is equal to the proportion of adult Chinese who pay with a credit or debit card. Mathematically, \(H_0: p_1 = p_2\). - Alternative hypothesis (H1): The proportion of adult Americans who pay with a credit or debit card is greater than the proportion of adult Chinese who pay with a credit or debit card. Mathematically, \(H_1: p_1 > p_2\), where \(p_1\) is the proportion of adult Americans and \(p_2\) is the proportion of adult Chinese.

02

Check if the samples are large enough

To determine if the samples are large enough for the large-sample test for a difference in population proportions, we need to check whether the following conditions are met: - \(n_1p_1 \ge 10\) - \(n_1(1-p_1) \ge 10\) - \(n_2p_2 \ge 10\) - \(n_2(1-p_2) \ge 10\) We are given the sample sizes and proportions as follows: - For the United States: \(n_1 = 1000\) and \(p_1 = 0.63\) - For China: \(n_2 = 1000\) and \(p_2 = 0.34\) Let's check the conditions: - \(1000 \times 0.63 \ge 10\) -> \(630 \ge 10\) (True) - \(1000 \times (1 - 0.63) \ge 10\) -> \(370 \ge 10\) (True) - \(1000 \times 0.34 \ge 10\) -> \(340 \ge 10\) (True) - \(1000 \times (1 - 0.34) \ge 10\) -> \(660 \ge 10\) (True) All the conditions are met, so the samples are large enough for the large-sample test for a difference in population proportions to be appropriate.

03

Calculate the test statistic and associated P-value

Follow the given Minitab output to calculate the value of the test statistic and the associated P-value.

04

Make a decision to reject or fail to reject the null hypothesis

We are given a significance level of 0.01. If the calculated P-value from the Minitab output is less than the significance level, we will reject the null hypothesis. If the P-value is greater than the significance level, we will fail to reject the null hypothesis.

05

Interpret the result of the hypothesis test

After determining whether to reject or fail to reject the null hypothesis, interpret the result in the context of the problem. This will inform us if there is convincing evidence suggesting that the proportion of adult Americans who pay with a credit card or a debit card is greater than the proportion of adult Chinese who pay with a credit or a debit card for online purchases.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Proportions

Let's dive into the idea of population proportions, which are essential when comparing different groups. In this context, population proportion refers to the fraction of individuals in a group who exhibit a particular trait or behavior. For example, in the exercise, it's the portion of Americans and Chinese using credit or debit cards for online purchases.

Calculating these proportions helps us understand and compare behaviors across two populations.

• For the United States: The proportion (\(p_1\)) is 0.63, indicating that 63% of the sample uses cards.
• For China: The proportion (\(p_2\)) is 0.34, meaning 34% use cards.

These proportions become the baseline for hypotheses testing, allowing us to see if the observed differences in behaviors between two groups are statistically significant or just due to random chance.

Sample Size

Sample size plays a crucial role in hypothesis testing as it directly affects the reliability of our conclusions. Larger sample sizes generally provide more accurate estimates of the population parameters.

In hypothesis testing for population proportions, we stress that the sample needs to be large enough to meet certain conditions, ensuring that the test is valid. These conditions are:

• \(n_1p_1 \ge 10\): Verification for the U.S. sample.
• \(n_1(1-p_1) \ge 10\): Another check for the U.S.
• \(n_2p_2 \ge 10\): Verification for the Chinese sample.
• \(n_2(1-p_2) \ge 10\): Another check for China.

In this exercise, both the American and Chinese samples have 1000 respondents each. As demonstrated, both meet the necessary conditions, allowing for a large-sample test which makes our results more dependable.

Significance Level

The significance level, often denoted as \(\alpha\), is a threshold set by the researcher that determines the probability of rejecting the null hypothesis when it is actually true. It's a way to control for Type I errors, which are false positives in hypothesis testing.

In this exercise, the significance level is chosen to be 0.01, meaning there is a 1% risk of incorrectly concluding that the proportion of Americans using cards is greater than that of the Chinese, when in fact it might not be true. This low significance level signifies a stricter criterion, reducing the chance of a false positive.

Significance levels play a vital role in determining the reliability of our hypothesis test results. Understanding it helps with making informed decisions about the null hypothesis.

Alternative Hypothesis

The alternative hypothesis is a key component of hypothesis testing. It specifies what we aim to prove and stands in opposition to the null hypothesis. For instance, it aligns with our belief or assumption that the actual state is different from what's stated under the null hypothesis.

In this problem, the alternative hypothesis is:

• \(H_1: p_1 > p_2\), suggesting that the proportion of Americans using credit or debit cards for online purchases is greater than that of the Chinese sample.

This hypothesis reflects our research question—if American shoppers truly use cards more frequently than their Chinese counterparts for online payments.

Formulating a clear alternative hypothesis allows researchers to focus their test and draw specific conclusions about the populations being compared. It's vital because it guides the entire hypothesis testing process, leading towards significant inferences.