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Multiple choice: Select the best answer for Exercises 29 to 32. A sample survey interviews SRSs of 500 female college students and 550 male college students. Each student is asked whether he or she worked for pay last summer. In all, 410 of the women and 484 of the men say 鈥淵es.鈥� Exercises 29 to 31 are based on this survey. Take \(\rho_{M}\) and \(p_{F}\) to be the proportions of all college males and females who worked last summer. We conjectured before seeing the data that men are more likely to work. The hypotheses to be tested are (a) \(H_{0} : p_{M}-p_{F}=0\) versus \(H_{a} : p_{M}-p_{F} \neq 0\) (b) \(H_{0} : p_{M}-p_{F}=0\) versus \(H_{a} : p_{M}-p_{F}>0\) (c) \(H_{0} : p_{M}-p_{P}=0\) versus \(H_{a} : p_{M}-p_{F}<0\) (d) \(H_{0} : p_{M}-p_{P}>0\) versus \(H_{a} : p_{M}-p_{P}=0\) (e) \(H_{0} : p_{M}-p_{F} \neq 0\) versus \(H_{a} : p_{M}-p_{F}=0\)

Step by step solution

01

Identify the Hypothesis Type

We need to select the hypothesis that fits the condition mentioned in the problem. The question states that it is conjectured that men are more likely to work, which is an assumption that leads to a one-sided test, looking for if men are more likely to work (greater proportion of workers).

02

Analyze the Hypotheses

We need to examine each of the provided options: - Option (a) is a two-sided test. - Option (b) represents a one-sided test with the alternative hypothesis that the proportion of men who worked is greater than the proportion of women. - Option (c) suggests a one-sided test that men are less likely to work. - Option (d) has the hypotheses reversed in inequality and is incorrectly formed. - Option (e) reverses the hypothesis and is incorrectly formed.

03

Choose the Appropriate Hypothesis

Based on the problem statement of conjecturing that men are more likely to work, and understanding each hypothesis, the correct option is (b) where the null hypothesis is that the difference is zero and the alternative hypothesis is that the proportion of men who worked is greater than the proportion of women.

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Key Concepts

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

Sample Survey

A sample survey is a method used to gather information from a subset, or sample, of a larger population. The process involves selecting participants from the larger group and asking them questions to gain insights about the entire population. In this context, the survey interviews 500 female and 550 male college students. The purpose of this survey is to understand their employment status during the summer. Performing a sample survey has several advantages:

• It offers a cost-effective way to gather data compared to surveying the entire population.
• It provides estimates that are usually accurate enough to provide useful insights.
• It is quicker than a full population study.

For credible results, it's crucial to ensure that the survey sample is representative of the population. For instance, using SRS or Simple Random Sampling aims to have every individual in the population have an equal chance of being chosen, minimizing biases and enhancing the reliability of the survey outcomes.

Proportions

Proportions in statistics are used to express the relative size of a part compared to the whole. It is especially useful when summarizing data related to two categories or classifications. In our exercise, we look at the proportion of college students who worked over the summer out of the total number interviewed.The proportion for a specific group is calculated using the formula:\[p = \frac{x}{n}\]where:

• \(x\) is the number of successes or favorable outcomes (e.g., number of students who worked).
• \(n\) is the total number of observations (e.g., total number of students surveyed).

For the sample, 410 of 500 women worked, giving a proportion of \(\frac{410}{500} = 0.82\). Similarly, 484 of 550 men worked, resulting in a proportion of \(\frac{484}{550} = 0.88\). These proportions help in comparing if one group is more likely to work than the other.

One-sided Test

A one-sided test in hypothesis testing is used when the research question or hypothesis predicts a specific direction of the effect. It tests if a parameter is greater than or less than a certain value, not simply unequal. In the given exercise, the hypothesis is that men are more likely to work compared to women. Thus, we use a one-sided test to investigate if the male proportion is significantly larger than the female proportion.The one-sided nature of the test influences both the formulation of the hypothesis and the statistical methods used to evaluate it:

• Null Hypothesis (\(H_0\)): There is no difference (\(p_M - p_F = 0\)).
• Alternative Hypothesis (\(H_a\)): The proportion of men is greater than that of women (\(p_M - p_F > 0\)).

This setup allows for the detection of directional differences, specifically checking if men are more likely to work based only on the comparison where \(p_M > p_F\).

AP Statistics

Advanced Placement (AP) Statistics is a comprehensive course designed to engage high school students in a wide variety of statistical concepts, such as sampling methods, descriptive statistics, probability, and inferential statistics like hypothesis testing. The course prepares students to tackle real-world problems using statistical techniques, much like the survey and hypothesis test problem we're analyzing. Key aspects of AP Statistics include:

• Learning to design studies and collect data responsibly, ensuring representative samples through methods like Simple Random Sampling.
• Developing skills to summarize and interpret data using graphical and numerical methods.
• Understanding how to use statistical techniques to make inferences about populations based on sample data.
• Becoming proficient in selecting and using proper statistical tests, such as one-sided or two-sided tests, depending on the context of the research question or hypothesis.

The hypothesis test problem illustrates how AP Statistics builds a foundation for understanding how to explore data and make reasoned conclusions, which is invaluable for further academic and professional pursuits in fields that rely on data-driven decision-making.