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Chapter 8: Problem 93

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.

Short Answer

Expert verified
Given the significance level and the calculated p-value, if the p-value is less than or equal to 0.05, we would conclude that there is evidence to suggest that the satisfaction with the environment among Americans has decreased. Otherwise, we would conclude that there is not enough evidence to suggest a decrease in satisfaction.

Step by step solution

01

State the Hypotheses

The Null Hypothesis, \(H_0\), is that the proportion is equal to \(0.52\), which is to say, the satisfaction has not decreased. The Alternative Hypothesis, \(H_1\), is that the proportion is less than \(0.52\), suggesting that the satisfaction has decreased. Mathematically, this can be represented as: \(H_0: p = 0.52\) and \(H_1: p < 0.52\)
02

Calculate the Test Statistic

For a hypothesis test of a population proportion, the test statistic is a Z-score (Z). It can be calculated using the formula: \(Z = (p虃 - p) / \sqrt{p(1 - p)/n}\). Here, \(p虃\) is the sample proportion, \(p\) is the hypothesized population proportion (from \(H_0\)), and \(n\) is the sample size. The sample proportion can be calculated as 461/1024 = 0.45. So, the test statistic Z which equals to \((0.45 - 0.52) / \sqrt{(0.52*0.48)/1024}\), need to be calculated by this formula
03

Find the P-value

The P-value is the probability that the test statistic will take a value as extreme as or more extreme than the observed value, assuming that the null hypothesis is true. This can be obtained from the standard normal (Z-) distribution table, or by using statistical software. For practical purposes, if the calculated p-value is less than the significance level (0.05), then reject the null hypothesis.
04

Make a Decision

Based on the p-value comparison with the given significance level, if the p-value is less than or equal to 0.05, we reject the null hypothesis and conclude that there is evidence to suggest that the satisfaction has decreased. Otherwise, we do not reject the null hypothesis and we conclude that there is not enough evidence to suggest a decrease in satisfaction.

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Most popular questions from this chapter

A college chemistry instructor thinks the use of embedded tutors will improve the success rate in introductory chemistry courses. The passing rate for introductory chemistry is \(62 \%\). During one semester, 200 students were enrolled in introductory chemistry courses with an embedded tutor. Of these 200 students, 140 passed the course. a. What is \(\hat{p}\), the sample proportion of students who passed introductory chemistry. b. What is \(p_{0}\), the proportion of students who pass introductory chemistry if the null hypothesis is true? c. Find the value of the test statistic. Explain the test statistic in context.

Give the null and alternative hypotheses for each test, and state whether a one-proportion z-test or a two-proportion z-test would be appropriate. a. You test a person to see whether he can tell tap water from bottled water. You give him 20 sips selected randomly (half from tap water and half from bottled water) and record the proportion he gets correct to test the hypothesis. b. You test a random sample of students at your college who stand on one foot with their eyes closed and determine who can stand for at least 10 seconds, comparing athletes and nonathletes.

The mother of a teenager has heard a claim that \(25 \%\) of teenagers who drive and use a cell phone reported texting while driving. She thinks that this rate is too high and wants to test the hypothesis that fewer than \(25 \%\) of these drivers have texted while driving. Her alternative hypothesis is that the percentage of teenagers who have texted when driving is less than \(25 \%\).$$\begin{aligned} &\mathrm{H}_{0}: p=0.25 \\\&\mathrm{H}_{\mathrm{a}}: p<0.25\end{aligned}$$ She polls 40 randomly selected teenagers, and 5 of them report having texted while driving, a proportion of \(0.125 .\) The p-value is \(0.034\). Explain the meaning of the p-value in the context of this question.

If we reject the null hypothesis, can we claim to have proved that the null hypothesis is false? Why or why not?

When a person stands trial for murder, the jury is instructed to assume that the defendant is innocent. Is this claim of innocence an example of a null hypothesis, or is it an example of an alternative hypothesis?
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