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Chapter 8: Problem 15

In 2016 the Harris poll estimated that \(3.3 \%\) of American adults are vegetarian. A nutritionist thinks this rate has increased. The nutritionist samples 150 American adults and finds that 11 are vegetarian. a. What is \(\hat{p}\), the sample proportion of vegetarians? b. What is \(p_{0}\), the hypothetical proportion of vegetarians? c. Find the value of the test statistic. Explain the test statistic in context.

Short Answer

Expert verified
The sample proportion of vegetarians \(\hat{p}\) is \(\frac{11}{150}\), the hypothetical proportion \(p_0\) is \(0.033\), and the test statistic can be calculated using the formula \[Z = \frac{{\hat{p} - p_0}}{{\sqrt{\frac{{p_0 \cdot (1 - p_0)}}{n}}}}\]

Step by step solution

01

Find the Sample Proportion

To find the sample proportion \(\hat{p}\), divide the number of successes (vegetarians) by the total sample size. In this case, \(\hat{p}\) = \(\frac{11}{150}\)
02

Identify the Hypothetical Proportion

The hypothetical proportion \(p_0\) is given in the problem as the estimated proportion of vegetarians in 2016, which is \(3.3%\). To convert this to a proportion in decimal form, divide by 100, so \(p_0\) = \(0.033\)
03

Compute the Test Statistic

Substitute the values \(\hat{p}\), \(p_0\), and \(n = 150\) into the formula in the above to calculate the test statistic. This will give you the standardized measure of how far \(\hat p\) is from \(p_0\)

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Most popular questions from this chapter

In problem \(8.15\) the nutritionist was interested in knowing if the rate of vegetarianism in American adults has increased. She carried out a hypothesis test and found that the observed value of the test statistic was \(2.77 .\) We can calculate that the p-value associated with this is \(0.0028\), which is very close to 0\. Explain the meaning of the p-value in this context. Based on this result, should the nutritionist believe the null hypothesis is true?

According to a 2015 University of Michigan poll, \(71.5 \%\) of high school seniors in the United States had a driver's license. A sociologist thinks this rate has declined. The sociologist surveys 500 randomly selected high school seniors and finds that 350 have a driver's license. a. Pick the correct null hypothesis. i. \(p=0.715\) ii. \(p=0.70\) iii. \(\hat{p}=0.715\) iv. \(\hat{p}=0.70\) b. Pick the correct alternative hypothesis. i. \(p>0.715\) ii. \(p<0.715\) iii. \(\hat{p}<0.715\) iv. \(p \neq 0.715\) c. In this context, the symbol \(p\) represents (choose one) i. the proportion of high school seniors in the entire United States that have a driver's license. ii. the proportion of high school seniors in the sociologist's random sample that have a driver's license.

In problem \(8.16\), a college chemistry instructor thinks the use of embedded tutors will improve the succes: rate in introductory chemistry courses. The instructor carried out a hypothesis test and found that the observed value of the test statistic was \(2.33 .\) The \(\mathrm{p}\) -value associated with this test statistic is \(0.0099 .\) Explain the meaning of the p-value in this context. Based on this result, should the instructor believe the success rate has improved?

A student who claims that he can tell tap water from bottled water is blindly tested with 20 trials. At each trial, tap water or bottled water is randomly chosen and presented to the student who much correctly identify the type of water. The experiment is designed so that the student will have exactly 10 sips from each type of water. He gets 13 identifications right out of 20 . Can the student tell tap water from bottled water at a \(0.05\) level of significance? Explain.

When comparing two sample proportions with a two-sided alternative hypothesis, all other factors being equal, will you get a smaller p-value with a larger sample size or a smaller sample size? Explain.
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