91Ó°ÊÓ

In each case. choose whether the appropriate test is a one-proportion \(z\) -test or a two-proportion z-test. Name the population(s). a. A researcher takes a random sample of 4 -year-olds to find out whether girls or boys are more likely to know the alphabet. b. A pollster takes a random sample of all U.S. adult voters to see whether more than \(50 \%\) approve of the performance of the current U.S. president. c. A researcher wants to know whether a new heart medicine reduces the rate of heart attacks compared to an old medicine. d. A pollster takes a poll in Wyoming about homeschooling to find out whether the approval rate for men is equal to the approval rate for women. e. A person is studied to see whether he or she can predict the results of coin flips better than chance alone.

Step by step solution

01

Scenario a

For this question, we are comparing two different proportions (boys and girls). Hence, the appropriate test is two-proportion \(z\)-test. The populations in this scenario are all 4-year old girls and boys.

02

Scenario b

In this case, we are only checking a single proportion (whether U.S. adult voters' approval rate is more than 50 % or not). Therefore, the appropriate test is one-proportion \(z\)-test. The population here is all U.S adult voters.

03

Scenario c

Here, we are comparing two proportions (the rate of heart attacks with new medicine vs old medicine). Therefore the appropriate test to be used is two-proportion \(z\) test. The populations considered are all individuals taking the old and new heart medicine.

04

Scenario d

The question deals with comparison of two proportions i.e., the homeschooling approval rate for men and women. Hence, the test to be used is two-proportion \(z\) test. The populations are all men and women in Wyoming.

05

Scenario e

Since we're checking if a person's prediction is better than chance (50%), we're only dealing with one proportion. Therefore, the test to be used is one-proportion \(z\) test. The population here is the individual being studied.

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Key Concepts

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

One-proportion z-test

The one-proportion z-test is a statistical method used to determine if a sample proportion significantly differs from a known or hypothesized population proportion. This test is particularly useful when you want to compare a single sample against a benchmark. In terms of application, consider a scenario where a pollster wants to find out if more than 50% of voters approve of a president's performance. The sample of voters is compared against the hypothesized proportion of 50% approval.

• Hypothesis Setup: Establish the null hypothesis (e.g., the true proportion equals the hypothesized value) and the alternative hypothesis (e.g., the true proportion does not equal the hypothesized value).
• Calculation: The z-score is calculated using the formula \( z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \), where \( \hat{p} \) is the sample proportion, \( p_0 \) is the hypothesized population proportion, and \( n \) is the sample size.
• Result Interpretation: Compare the calculated z-score to the critical z-value from the z-table to decide whether to reject the null hypothesis.

Given this understanding, the one-proportion z-test is optimal when dealing with questions that confront a single proportion against a standard value.

Two-proportion z-test

The two-proportion z-test extends the principles of the one-proportion z-test by comparing two independent sample proportions to see if there is a significant difference between them. This test is applicable in situations like determining whether boys or girls are more likely to know the alphabet, or comparing the effectiveness of two medical treatments.

• Hypothesis Setup: The null hypothesis posits that the two proportions are equal, while the alternative hypothesis states that they are not.
• Calculation: The formula for the z-score in this test is \( z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}} \), where \( \hat{p}_1 \) and \( \hat{p}_2 \) are the sample proportions, and \( \hat{p} \) is the pooled sample proportion.
• Result Interpretation: Compare the z-score against the critical values to determine if there is enough evidence to reject the null hypothesis.

This test is best used when your research or analysis focuses on comparing two distinct groups to assess differences in proportions.

Statistical Comparison

Statistical comparison encompasses the methods and tools used to compare different statistical data points. These comparisons help identify significant differences, trends, or relationships within datasets. Understanding statistical comparisons is crucial in hypothesis testing, where the aim is often to determine if there is a significant effect or difference present.

• Objective: The primary goal is to determine if observed differences are due to genuine variations or random chance.
• Tools: Tests such as one-proportion z-test or two-proportion z-test are employed based on the number of groups and nature of the data.
• Implementation: After setting up hypotheses, appropriate statistical tests are applied to analyze the data, leading to conclusions about the relationships or differences.

Through statistical comparisons, researchers can make informed decisions or observations that answer their specific questions.

Population Sampling

Population sampling is a method used to select a subset of individuals from a total population to draw conclusions about that population. It is an essential component of statistical methods which help in making inferences about the larger group based on the smaller sample.

• Purpose: Sampling helps in gaining insights without the need for a complete census, saving time and resources.
• Types: Common approaches include random sampling, stratified sampling, and systematic sampling, among others.
• Application: Proper sampling is crucial for ensuring that the sample is representative of the population so that any findings can be generalized reliably.

In the context of z-tests, population sampling allows for the assumption that the sample represents the larger population, thus enabling valid hypothesis testing and conclusions.