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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

Choose one of the answers in each case. In statistical inference, measurements are made on a ______ (sample or population), and generalizations are made to a _____ (sample or population).

For each of the following, state whether a one-proportion \(z\) -test or a two- proportion \(z\) -test would be appropriate, and name the population(s). a. A polling agency takes a random sample of voters in California to determine if a ballot proposition will pass. b. A researcher asks a random sample of residents from coastal states and a random sample of residents of non-coastal states whether they favor increased offshore oil drilling. The researcher wants to determine if there is a difference in the proportion of residents who support off-shore drilling in the two regions.

St. Louis County is \(24 \%\) African American. Suppose you are looking at jury pools, each with 200 members, in St. Louis County. The null hypothesis is that the probability of an African American being selected into the jury pool is \(24 \%\). a. How many African Americans would you expect on a jury pool of 200 people if the null hypothesis is true? b. Suppose pool A contains 40 African American people out of 200 , and pool B contains 26 African American people out of 200 . Which will have a smaller p-value and why?

If we do not reject the null hypothesis, is it valid to say that we accept the null hypothesis? Why or why not?

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