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Chapter 10: Problem 27

Decide whether the problem requires a confidence interval or hypothesis test, and determine the variable of interest. For any problem requiring a confidence interval, state whether the confidence interval will be for a population proportion or population mean. For any problem requiring a hypothesis test, write the null and alternative hypothesis. In \(2014,\) of the 37 million borrowers who have outstanding student loan balances, \(14 \%\) have at least one past due student loan account. A researcher with the United States Department of Education believes this proportion has increased since then.

Short Answer

Expert verified
This problem requires a hypothesis test. The null hypothesis is H_0: p = 0.14 and the alternative hypothesis is H_a: p > 0.14.

Step by step solution

01

Identify the Type of Problem

Determine if the problem requires a confidence interval or a hypothesis test. The problem states that a researcher believes the proportion of past due student loan accounts has increased since 2014.
02

Determine the Variable of Interest

The variable of interest is the proportion of borrowers having at least one past due student loan account.
03

Justify the Need for a Hypothesis Test

Since the researcher believes there has been a change (specifically an increase) in the proportion, a hypothesis test is needed to test this belief statistically.
04

State the Null Hypothesis (H_0)

The null hypothesis is the statement that there has been no change in the proportion of past due accounts: H_0: p = 0.14
05

State the Alternative Hypothesis (H_a)

The alternative hypothesis is the statement that the proportion of past due accounts has increased: H_a: p > 0.14

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Key Concepts

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

confidence interval
A confidence interval gives us a range where we believe the true population parameter lies.
It is commonly used to estimate population means or proportions.
For instance, if we want to know the proportion of students with past due loans, we can calculate a confidence interval to see the plausible range for this proportion.
We use confidence levels like 95% to express our degrees of certainty in this range.
This method provides more information compared to point estimates
since it offers a span of values rather than a single number.
population proportion
Population proportion refers to the fraction of individuals in a population that have a particular characteristic.
In the given example, it refers to the proportion of borrowers with at least one past due student loan account.
We often represent this proportion with the symbol \( p \).
To estimate it accurately, we need to collect sample data and apply statistical techniques.
Confidence intervals and hypothesis testing are two primary methods for analyzing population proportions.
These methods help us to make inferences about the broader population beyond just the sample data.
null hypothesis
The null hypothesis (\( H_0 \)) is a fundamental concept in statistical analysis.
It is a statement that proposes no effect or no difference, and it assumes that any observed outcomes are due to chance. In the example, \( H_0 \) is stated as \( p = 0.14 \), meaning there has been no change in the proportion of past due student loan accounts since 2014.
The null hypothesis serves as the default or starting assumption in hypothesis testing.
We collect data and perform analyses to see if there is sufficient evidence to reject the null hypothesis in favor of an alternative hypothesis.
alternative hypothesis
The alternative hypothesis (\( H_a \)) contradicts the null hypothesis.
It suggests that there is an actual effect or difference.
For our example, the alternative hypothesis is \( H_a: p > 0.14 \), indicating that the researcher believes the proportion of past due accounts has increased since 2014.
When conducting a hypothesis test, we analyze the data to determine whether we have enough evidence to reject the null hypothesis in favor of the alternative.
The alternative hypothesis helps to direct our investigation and clarifies what we are trying to prove with our data.
statistical analysis
Statistical analysis is a set of methods used to collect, review, analyze, and draw conclusions from data.
It encompasses various techniques to handle different types of data and hypotheses.
In the given problem, statistical analysis helps us determine if there is evidence to support the researcher's claim.
We decide whether to use confidence intervals or hypothesis tests depending on the nature of the question.
  • Confidence intervals estimate the range where the population parameter lies,
  • while hypothesis tests help us make decisions about competing hypotheses.
Proper statistical analysis ensures that our conclusions are data-driven and reliable.

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Most popular questions from this chapter

Test the hypothesis using (a) the classical approach and (b) the P-value approach. Be sure to verify the requirements of the test. $$\begin{array}{l}H_{0}: p=0.6 \text { versus } H_{1}: p<0.6 \\\n=250 ; x=124 ; \alpha=0.01\end{array}$$

The manufacturer of Prolong Engine Treatment claims that if you add one 12 -ounce bottle of its \(\$ 20\) product, your engine will be protected from excessive wear. An infomercial claims that a woman drove 4 hours without oil, thanks to Prolong. Consumer Reports magazine tested engines in which they added Prolong to the motor oil, ran the engines, drained the oil, and then determined the time until the engines seized. (a) Determine the null and alternative hypotheses that Consumer Reports will test. (b) Both engines took exactly 13 minutes to seize. What conclusion might Consumer Reports draw based on this evidence?

Filling Bottles A certain brand of apple juice is supposed to have 64 ounces of juice. Because the penalty for underfilling bottles is severe, the target mean amount of juice is 64.05 ounces. However, the filling machine is not precise, and the exact amount of juice varies from bottle to bottle. The quality-control manager wishes to verify that the mean amount of juice in each bottle is 64.05 ounces so that she can be sure that the machine is not over- or underfilling. She randomly samples 22 bottles of juice, measures the content, and obtains the following data: $$ \begin{array}{llllll} \hline 64.05 & 64.05 & 64.03 & 63.97 & 63.95 & 64.02 \\ \hline 64.01 & 63.99 & 64.00 & 64.01 & 64.06 & 63.94 \\ \hline 63.98 & 64.05 & 63.95 & 64.01 & 64.08 & 64.01 \\ \hline 63.95 & 63.97 & 64.10 & 63.98 & & \\ \hline \end{array} $$ A normal probability plot suggests the data could come from a population that is normally distributed. A boxplot does not show any outliers. (a) Should the assembly line be shut down so that the machine can be recalibrated? Use a 0.01 level of significance. (b) Explain why a level of significance of \(\alpha=0.01\) is more reasonable than \(\alpha=0.1 .\) [Hint: Consider the consequences of incorrectly rejecting the null hypothesis.

According to menstuff.org, \(22 \%\) of married men have "strayed" at least once during their married lives. (a) Describe how you might go about administering a survey to assess the accuracy of this statement. (b) A survey of 500 married men indicated that 122 have "strayed" at least once during their married life. Construct a \(95 \%\) confidence interval for the population proportion of married men who have strayed. Use this interval to assess the accuracy of the statement made by menstuff.org.

If we do not reject the null hypothesis when the statement in the alternative hypothesis is true, we have made a Type ____ error.
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