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Chapter 8: Problem 25

In a Northeastern University/Gallup poll of 461 young Americans aged 18 to 35,152 reported they would be comfortable riding in a self-driving car. Suppose we are testing the hypothesis that more than \(30 \%\) of Americans in this age group would be comfortable riding in a self-driving car, using a significance level of \(0.05 .\) Which of the following figures correctly matches the alternative hypothesis \(p>0.30 .\) Report and interpret the correct p-value.

Short Answer

Expert verified
The short answer will be obtained at the end of Step 4 and depends on the calculated p-value. If the p-value is less than 0.05, the short answer would be 'We reject the null hypothesis and conclude that more than 30% of young Americans are comfortable riding in a self-driving car', otherwise the short answer would be 'We fail to reject the null hypothesis and conclude that we don't have sufficient evidence to say that more than 30% of young Americans are comfortable riding in a self-driving car'.

Step by step solution

01

Formulate Hypotheses

The null hypothesis \(H_0\) is that the proportion is 30% or less, i.e., \(p \leq 0.30\). The alternative hypothesis \(H_a\) is that the proportion is greater than 30%, i.e., \(p > 0.30\).
02

Calculate the Test Statistic

We use the formula for the z-score, which is a test statistic: \(z = \frac{\hat{p} - p_0}{\sqrt{p_0(1-p_0)/n}}\). Here, \(\hat{p}\) is the sample proportion, \(p_0\) is the proportion under null hypothesis, and n is the sample size. Substitute \(\hat{p} = \frac{152}{461}\), \(p_0 = 0.30\), and \(n = 461\) into the formula to find the z-score.
03

Calculate the P-value

Using a Z-table or appropriate statistical software, find the probability that the Z is larger than the calculated z-score. This is the p-value.
04

Make a Decision

If the p-value is less than the significance level of 0.05, reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
05

Interpret the Results

State the conclusion in the context of the problem. If the null hypothesis was rejected, it implies that a statistically significant proportion of young Americans are comfortable riding in a self-driving car.

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Most popular questions from this chapter

A researcher carried out a hypothesis test using a two-sided alternative hypothesis. Which of the following z-scores is associated with the smallest p-value? Explain. i. \(z=0.50\) ii. \(z=1.00\) iii. \(z=2.00 \quad\) iv. \(z=3.00\)

A multiple-choice test has 50 questions with four possible options for each question. For each question, only one of the four options is correct. A passing grade is 35 or more correct answers. a. What is the probability that a person will guess correctly on one multiple- choice question? b. Test the hypothesis that a person who got 35 right out of 50 is not just guessing, using an alpha of \(0.05\). Steps 1 and 2 of the hypothesis testing procedure are given. Finish the question by doing steps 3 and 4 . Step 1: \(\quad \mathrm{H}_{0}: p=0.25\) \(\mathrm{H}_{\mathrm{a}}: p>0.25\) Step 2: Choose the one-proportion \(z\) -test. \(n\) times \(p\) is 50 times \(0.25\), which is \(12.5\). This is more than 10 , and 50 times \(0.75\) is also more than 10 . Assume a random sample.

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