/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 41 The report "Audience Insights: C... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 41

The report "Audience Insights: Communicating to Teens (Aged 12-17)" (www.cdc.gov, 2009) described teens' attitudes about traditional media, such as TV, movies, and newspapers. In a representative sample of American teenage girls, \(41 \%\) said newspapers were boring. In a representative sample of American teenage boys, \(44 \%\) said newspapers were boring. Sample sizes were not given in the report. a. Suppose that the percentages reported had been based on a sample of 58 girls and 41 boys. Is there convincing evidence that the proportion of those who think that newspapers are boring is different for teenage girls and boys? Carry out a hypothesis test using \(\alpha=.05\) b. Suppose that the percentages reported had been based on a sample of 2000 girls and 2500 boys. Is there convincing evidence that the proportion of those who think that newspapers are boring is different for teenage girls and boys? Carry out a hypothesis test using \(\alpha=.05\). c. Explain why the hypothesis tests in Parts (a) and (b) resulted in different conclusions.

Short Answer

Expert verified
The results of two hypothesis tests might be different due to the different sample sizes. Larger samples lead to more accurate estimation of the population, and could detect even a small difference between the two proportions, while smaller samples might fail to capture such a difference. Without the actual data, more specific conclusion can't be provided.

Step by step solution

01

State the Hypotheses

In both cases we're looking to see if there's a difference in proportion between two populations - teenage girls and boys. Therefore, the null hypothesis \(H_0: p_1 - p_2 = 0\), and the alternative hypothesis \(H_A: p_1 - p_2 \neq 0\). Where \(p_1\) is the proportion of girls who find newspapers boring and \(p_2\) is the proportion of boys who find newspapers boring.
02

Conduct the Hypothesis Tests

We carry out the first hypothesis test, using the sample sizes of 58 girls and 41 boys. From the information given, we know that \(x_1 = 0.41 * 58 = 24\) girls and \(x_2 = 0.44 * 41 = 18\) boys find newspapers boring. Then we calculate the sample proportions and their difference - \(\hat{p}_1 = x_1 / n_1\), \(\hat{p}_2 = x_2 / n_2\), and \(\hat{p} = x / n\) - and the standard error \(SE = \sqrt{\hat{p}*(1-\hat{p})*(1/n_1 + 1/n_2)}\). Then, we calculate the z-score \(z = (\hat{p}_1 - \hat{p}_2) / SE\). After that, we compute the p-value using a standard normal probability table. We conduct the similar steps for the second hypothesis test, using the sample sizes of 2000 girls and 2500 boys.
03

Decision and Conclusion

For each hypothesis test, if the computed p-value is less than \(\alpha = .05\), we reject the null hypothesis and conclude that there is a significant difference between teenage girls and boys in terms of finding newspapers boring. Otherwise, we fail to reject the null hypothesis. Then we compare the results from the two tests and explain why they differ, considering the impact of the sample sizes.

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.

Proportion Difference
In statistics, when comparing two groups, we are often interested in the difference between their proportions. This can tell us whether there's a notable deviation in outcomes, attitudes, or behaviors between the two groups.

In the context of the provided exercise, we calculate the proportion of teenage girls and boys who find newspapers boring and then determine the difference between these two proportions. If this difference is significantly different from zero, it may suggest that gender plays a role in the perception of newspapers. To find this proportion difference, we use the formula \( \hat{p}_1 - \hat{p}_2 \), where \( \hat{p}_1 \) represents the sample proportion of girls, and \( \hat{p}_2 \) represents the sample proportion of boys. Accurately determining the proportion difference is critical for effective hypothesis testing.
Null Hypothesis
The null hypothesis, symbolized as \( H_0 \), is a fundamental concept in hypothesis testing. It is a statement that there is no effect or no difference, and it serves as the starting assumption for statistical tests.

In both scenarios of our exercise, the null hypothesis posits that there is no difference in the proportion of girls and boys who believe newspapers are boring, formally stated as \( H_0: p_1 - p_2 = 0 \). We assume this to be true unless we find sufficient evidence to the contrary. The null hypothesis is what we attempt to reject through our hypothesis test, and it is a crucial step in determining the statistical significance of the observed data.
Alternative Hypothesis
The alternative hypothesis, denoted as \( H_A \), represents the opposite of the null hypothesis and indicates the presence of an effect or a difference. In simple terms, it's what we suspect might be true instead of the null hypothesis.

For the exercise, the alternative hypothesis states that there is a difference in the proportion of teenage girls and boys who find newspapers boring, indicated as \( H_A: p_1 - p_2 eq 0 \). It's the hypothesis that we are trying to support by using evidence from our sample data. The alternative hypothesis is what leads us to examine the p-value and assess whether our findings are statistically significant.
P-Value
The p-value is a vital term in hypothesis testing, representing the probability of obtaining a result at least as extreme as the one observed, given that the null hypothesis is true. In essence, it helps us determine the significance of our findings.

After calculating the test statistic鈥攐ften a z-score in the case of proportion differences鈥攚e look up this value in a standard normal distribution to find the corresponding p-value. In our exercise, if this p-value is less than our alpha level of \( \alpha = 0.05 \), we have strong evidence against the null hypothesis and can consider our findings significant. The lower the p-value, the higher the significance of our results in supporting the alternative hypothesis.
Statistical Significance
Statistical significance is the likelihood that the relationship or difference detected in a sample also exists in the wider population and was not due to chance. It provides assurance that our findings are meaningful and not just a fluke.

By using the threshold of \( \alpha = 0.05 \) in the exercise, we define a 5% risk as an acceptable level for making a Type I error鈥攖hat is, wrongly rejecting the null hypothesis. If the p-value is below this alpha level, our results are deemed statistically significant. A vital consideration鈥攚hich emerges in the exercise and its differing outcomes鈥攊s how sample size affects statistical significance. Larger samples tend to yield more reliable and detectable results, therefore enhancing statistical significance.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A researcher at the Medical College of Virginia conducted a study of 60 randomly selected male soccer players and concluded that frequently "heading" the ball in soccer lowers players' IQs (USA Today, August 14 1995). The soccer players were divided into two groups, based on whether they averaged 10 or more headers per game. Mean IQs were reported in the article, but the sample sizes and standard deviations were not given. Suppose that these values were as given in the accompanying table. $$ \begin{array}{l|ccc} & & \text { Sample } & \text { Sample } \\ & n & \text { Mean } & \text { sd } \\ \hline \text { Fewer Than 1O Headers } & 35 & 112 & 10 \\ 10 \text { or More Headers } & 25 & 103 & 8 \\ \hline \end{array} $$ Do these data support the researcher's conclusion? Test the relevant hypotheses using \(\alpha=.05 .\) Can you conclude that heading the ball causes lower \(\mathrm{IQ}\) ?

In a study of malpractice claims where a settlement had been reached, two random samples were selected: a random sample of 515 closed malpractice claims that were found not to involve medical errors and a random sample of 889 claims that were found to involve errors (New England Journal of Medicine [2006]: \(2024-2033\) ). The following statement appeared in the referenced paper: "When claims not involving errors were compensated, payments were significantly lower on average than were payments for claims involving errors \((\$ 313,205\) vs. \(\$ 521,560, P=0.004) . "\) a. What hypotheses must the researchers have tested in order to reach the stated conclusion? b. Which of the following could have been the value of the test statistic for the hypothesis test? Explain your reasoning. i. \(t=5.00\) ii. \(t=2.65\) iii. \(t=2.33\) iv. \(t=1.47\)

Consider two populations for which \(\mu_{1}=30\), \(\sigma_{1}=2, \mu_{2}=25,\) and \(\sigma_{2}=3 .\) Suppose that two independent random samples of sizes \(n_{1}=40\) and \(n_{2}=50\) are selected. Describe the approximate sampling distribution of \(\bar{x}_{1}-\bar{x}_{2}\) (center, spread, and shape).

Some commercial airplanes recirculate approximately \(50 \%\) of the cabin air in order to increase fuel efficiency. The authors of the paper "Aircraft Cabin Air Recirculation and Symptoms of the Common Cold" (Journal of the American Medical Association [2002]: \(483-486\) ) studied 1100 airline passengers who flew from San Francisco to Denver between January and April 1999\. Some passengers traveled on airplanes that recirculated air and others traveled on planes that did not recirculate air. Of the 517 passengers who flew on planes that did not recirculate air, 108 reported post-flight respiratory symptoms, while 111 of the 583 passengers on planes that did recirculate air reported such symptoms. Is there sufficient evidence to conclude that the proportion of passengers with post-flight respiratory symptoms differs for planes that do and do not recirculate air? Test the appropriate hypotheses using \(\alpha=.05\). You may assume that it is reasonable to regard these two samples as being independently selected and as representative of the two populations of interest.

Can moving their hands help children learn math? This question was investigated in the paper "Gesturing Gives Children New Ideas about Math" (Psychological Science [2009]: 267-272). Eighty-five children in the third and fourth grades who did not answer any questions correctly on a test with six problems of the form \(3+2+8=-8\) were participants in an experiment. The children were randomly assigned to either a no-gesture group or a gesture group. All the children were given a lesson on how to solve problems of this form using the strategy of trying to make both sides of the equation equal. Children in the gesture group were also taught to point to the first two numbers on the left side of the equation with the index and middle finger of one hand and then to point at the blank on the right side of the equation. This gesture was supposed to emphasize that grouping is involved in solving the problem. The children then practiced additional problems of this type. All children were then given a test with six problems to solve, and the number of correct answers was recorded for each child. Summary statistics read from a graph in the paper are given below. $$ \begin{array}{l|ccc} & n & \bar{x} & s \\ \hline \text { No gesture } & 42 & 1.3 & 0.3 \\ \text { Gesture } & 43 & 2.2 & 0.4 \\ \hline \end{array} $$ Is there evidence to support the theory that learning the gesturing approach to solving problems of this type results in a higher mean number of correct responses? Test the relevant hypotheses using \(\alpha=.01\).
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.