91影视

Teen Drivers According to a 2015 University of Michigan poll, \(71.5 \%\) of high school seniors in the United States had a driver's license. A sociologist thinks this rate has declined. The sociologist surveys 500 randomly selected high school seniors and finds that 350 have a driver's license. a. Pick the correct null hypothesis. i. \(p=0.715\) ii. \(p=0.70\) iii. \(\hat{p}=0.715\) iv. \(\hat{p}=0.70\) b. Pick the correct alternative hypothesis. i. \(p>0.715\) ii. \(p<0.715 \quad\) iii. \(\hat{p}<0.715 \quad\) iv. \(p \neq 0.715\) c. In this context, the symbol \(p\) represents (choose one) i. the proportion of high school seniors in the entire United States that have a driver's license. ii. the proportion of high school seniors in the sociologist's random sample that have a driver's license.

Step by step solution

01

Identify the Null Hypothesis

The null hypothesis is the initial claim that we assume to be true. It is about the population proportion (p), not the sample proportion (\(\hat{p}\)). So the null hypothesis here is \(p=0.715\).

02

Identify the Alternative Hypothesis

The alternative hypothesis is the claim we are testing against the null hypothesis. It is also about the population proportion, not the sample proportion. Since the sociologist thinks the rate has declined, the alternative hypothesis is \(p<0.715\).

03

Identifying the meaning of 'p'

The symbol 'p' represents the population proportion. Therefore, in this context, 'p' indicates the proportion of high school seniors in the entire United States that have a driver's license.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Null Hypothesis

The null hypothesis is a fundamental part of hypothesis testing. It represents an initial claim or a default position that there is no effect or difference. It is symbolized by the letter "H鈧�" and is about the population, not the sample.
In our exercise, the null hypothesis is that the true proportion of high school seniors in the United States with a driver's license is the same as reported before, specifically, 71.5%. This is expressed as:

• \( H_0: p = 0.715 \)

The null hypothesis aims to act as a benchmark for testing new claims and data, maintaining that nothing has changed unless evidence suggests otherwise. Rejecting the null hypothesis points to an alternative claim which can be confirmed through statistical testing.

Alternative Hypothesis

The alternative hypothesis offers a different perspective contrary to the null. It suggests that there is an effect or a difference, often represented as "H鈧�" or "H鈧�". This hypothesis is the reason a study or test is conducted, often proposing change or difference.
In the given scenario, the sociologist credibly believes that the percentage of high school seniors with a driver's license has diminished. Therefore, the alternative hypothesis to be considered is:

• \( H_1: p < 0.715 \)

This directional hypothesis specifically tests whether the current proportion is less than the original claim of 71.5%, hence it's a one-tailed test. Testing the alternative hypothesis involves statistical methods to determine if the observed data provides enough evidence to support this claim.

Population Proportion

The population proportion is a crucial component in hypothesis testing. It informs us about the characteristic of interest across a whole population which, in this context, is high school seniors in the entire United States with a driver's license. This is denoted by "p".
Population proportion gives an overview of the actual or hypothesized prevalence of an attribute in a large set of data.

• In our example, the population proportion is \( p = 0.715 \), referring to the 2015 data.

Conducting hypothesis testing revolves around making inferences about this proportion based on the sample data. Moreover, being clear about what "p" represents helps in setting the correct hypotheses and subsequent testing process.

Sample Proportion

The sample proportion, represented by "\(\hat{p}\)", is the estimate of the population proportion calculated from a randomly selected sample.
In hypothesis testing, the sample proportion helps to provide the data against the null hypothesis.
In our exercise, the sociologist surveyed 500 high school seniors and found that 350 of them had a driver's license, leading to a sample proportion calculated as follows:

• \( \hat{p} = \frac{350}{500} = 0.70 \)

Sample proportion acts as an estimator and guide for making inferences about the entire population's proportion. It is pivotal in statistical analysis and in helping determine if the observed variations are due to chance or signify an actual change from the presumed population proportion.