/*! 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 35 A 2018 Gallup poll of 3635 rando... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 35

A 2018 Gallup poll of 3635 randomly selected Facebook users found that 2472 get most of their news about world events on Facebook. Research done in 2013 found that only \(47 \%\) of all Facebook users reported getting their news about world events on Facebook. See page 430 for guidance. a. Does this sample give evidence that the proportion of Facebook users who get their world news on Facebook has changed since 2013 ? Carry out a hypothesis test and use a \(0.05\) significance level. b. After conducting the hypothesis test, a further question one might ask is what proportion of all Facebook users got most of their news about world events on Facebook in 2018 . Use the sample data to construct a \(90 \%\) confidence interval for the population proportion. How does your confidence interval support your hypothesis test conclusion?

Short Answer

Expert verified
The detailed answer would depend on the test statistic, P-value and confidence interval that we obtain in the calculations. However, the logic for solving this problem has been outlined above.

Step by step solution

01

State the Null and Alternative Hypotheses

The null hypothesis, denoted \(H_0\), is a statement of no effect or no difference. The alternative hypothesis, denoted \(H_a\), is a statement that indicates the presence of an effect or difference. For this problem, the null hypothesis (\(H_0\)) is that there is no change in the proportion of Facebook users who got most of their news on Facebook from 2013 to 2018, which means the proportion is \(47\%\). The alternative hypothesis (\(H_a\)) is that the proportion has changed, meaning it is not \(47\%\). Therefore, \(H_0: p = 0.47\) and \(H_a: p \neq 0.47\)
02

Compute the Test Statistic

For hypothesis testing, we use the formula for the Z statistic: \(Z = (p虃 - p_0) / \sqrt{(p_0(1-p_0) / n)}\), where \(p_0\) is the proportion according to the null hypothesis, \(p虃\) is the sample proportion, and \(n\) is the sample size. The sample proportion is calculated as \(p虃 = x / n = 2472 / 3635 \approx 0.68\), where \(x\) is the number of successes in the sample. Substituting the values in the formula, we calculate the test statistic.
03

Find the P-value

The P-value is the probability that, assuming the null hypothesis is true, the test statistic is as extreme as, or more extreme than, the observed test statistic. The P-value needs to be compared with the significance level (0.05) to make a decision about the hypotheses. If the P-value is less than or equal to the significance level, the null hypothesis can be rejected.
04

Draw Conclusion

Based on the comparison of the P-value and the significance level, a decision is made to either reject or fail to reject the null hypothesis.
05

Construct a 90% Confidence Interval

The confidence interval for a population proportion is calculated by: \(p虃 \pm z*(\sqrt{(p虃(1-p虃) / n)})\). Where \(z*\) is the z-score for the desired level of confidence, here, for a 90% confidence level, \(z* = 1.645\). From this, we obtain the confidence interval.
06

Interpret the Confidence Interval

If the confidence interval includes 0.47, then it supports the claim that there is no significant difference between the 2013 and 2018 proportions. Otherwise, it would suggest that there has been a significant change in the proportion of Facebook users getting their news on Facebook.

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!

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 true/false test has 50 questions. Suppose a passing grade is 35 or more correct answers. Test the claim that a student knows more than half of the answers and is not just guessing. Assume the student gets 35 answers correct out of \(50 .\) Use a significance level of \(0.05 .\) Steps 1 and 2 of a hypothesis test procedure are given. Show steps 3 and 4, and be sure to write a clear conclusion. Step $$\text { 1: } \begin{aligned}&\mathrm{H}_{0}: p=0.50 \\\&\mathrm{H}_{\mathrm{a}}: p>0.50\end{aligned}$$ Step 2: Choose the one-proportion \(z\) -test. Sample size is large enough, because \(n p_{0}\) is \(50(0.5)=25\) and \(n\left(1-p_{0}\right)=50(0.50)=25\), and both are more than \(10 .\) Assume the sample is random and \(\alpha=0.05\).

In 2015 a Gallup poll reported that \(52 \%\) of Americans were satisfied with the quality of the environment. In 2018 , a survey of 1024 Americans found that 461 were satisfied with the quality of the environment. Does this survey provide evidence that satisfaction with the quality of the environment among Americans has decreased? Use a \(0.05\) significance level.

According to a 2016 report by the Census Bureau, \(60.1 \%\) of women and \(57.6 \%\) of men have completed some college education or higher. Would it be appropriate to do a two-proportion z-test to determine whether the proportions of men and women who had completed some college education or higher were different (assuming we knew the total number of men and women)? Why or why not?

A 2018 Gallup poll of 2228 randomly selected U.S. adults found that \(39 \%\) planned to watch at least a "fair amount" of the 2018 Winter Olympics. In \(2014,46 \%\) of U.S. adults reported planning to watch at least a "fair amount." a. Does this sample give evidence that the proportion of U.S. adults who planned to watch the 2018 Winter Olympics was less than the proportion who planned to do so in 2014 ? Use a \(0.05\) significance level. b. After conducting the hypothesis test, a further question one might ask is what proportion of all U.S. adults planned to watch at least a "fair amount" of the 2018 Winter Olympics. Use the sample data to construct a \(90 \%\) confidence interval for the population proportion. How does your confidence interval support your hypothesis test conclusion?

A 20-question multiple choice quiz has five choices for each question. Suppose that a student just guesses, hoping to get a high score. The teacher carries out a hypothesis test to determine whether the student was just guessing. The null hypothesis is \(p=0.20\), where \(p\) is the probability of a correct answer. a. Which of the following describes the value of the \(z\) -test statistic that is likely to result? Explain your choice. i. The \(z\) -test statistic will be close to 0 . ii. the \(z\) -test statistic will be far from 0 . b. Which of the following describes the \(\mathrm{p}\) -value that is likely to result? Explain your choice. i. The p-value will be small. ii. The p-value will not be small.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Probability and Statistics

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.