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Chapter 4: Problem 11

A situation is described for a statistical test. In each case, define the relevant parameter(s) and state the null and alternative hypotheses. Testing to see if there is evidence that the percentage of a population who watch the Home Shopping Network is less than \(20 \%\).

Short Answer

Expert verified
The parameter of interest is \(p\), the proportion of the population who watch the Home Shopping Network. The null hypothesis is \(H_0: p \geq 0.20\) and the alternative hypothesis is \(H_a: p < 0.20\).

Step by step solution

01

Define the Parameter

In this case, we define the parameter of interest as \(p\), the proportion or percentage of the population who watch the Home Shopping Network.
02

State the Null Hypothesis

The Null Hypothesis, \(H_0\), is generally a statement of no difference or no effect. Here, we are testing if the percentage is less than 20%. So, our Null Hypothesis is the opposite of this, i.e., that the percentage is NOT less than 20 %. Thus, we have the null hypothesis \(H_0: p \geq 0.20\). That is, the proportion, \(p\), who watch the Home Shopping Network is equal to or greater than 20 %.
03

State the Alternative Hypothesis

The Alternative Hypothesis, \(H_a\), is a claim about the population that we are trying to find evidence for. Our task in this exercise is to test if the percentage is less than 20 %. Therefore, the alternative hypothesis is: \(H_a: p < 0.20\). This states that the proportion, \(p\), who watch the Home Shopping Network is less than 20 %.

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Most popular questions from this chapter

A study suggests that exposure to UV rays through the car window may increase the risk of skin cancer. \(^{52}\) The study reviewed the records of all 1,050 skin cancer patients referred to the St. Louis University Cancer Center in 2004\. Of the 42 patients with melanoma, the cancer occurred on the left side of the body in 31 patients and on the right side in the other 11 . (a) Is this an experiment or an observational study? (b) Of the patients with melanoma, what proportion had the cancer on the left side? (c) A bootstrap \(95 \%\) confidence interval for the proportion of melanomas occurring on the left is 0.579 to \(0.861 .\) Clearly interpret the confidence interval in the context of the problem. (d) Suppose the question of interest is whether melanomas are more likely to occur on the left side than on the right. State the null and alternative hypotheses. (e) Is this a one-tailed or two-tailed test? (f) Use the confidence interval given in part (c) to predict the results of the hypothesis test in part (d). Explain your reasoning. (g) A randomization distribution gives the p-value as 0.003 for testing the hypotheses given in part (d). What is the conclusion of the test in the context of this study? (h) The authors hypothesize that skin cancers are more prevalent on the left because of the sunlight coming in through car windows. (Windows protect against UVB rays but not UVA rays.) Do the data in this study support a conclusion that more melanomas occur on the left side because of increased exposure to sunlight on that side for drivers?

Exercise 4.19 on page 269 describes a study investigating the effects of exercise on cognitive function. \({ }^{31}\) Separate groups of mice were exposed to running wheels for \(0,2,4,7,\) or 10 days. Cognitive function was measured by \(Y\) maze performance. The study was testing whether exercise improves brain function, whether exercise reduces levels of BMP (a protein which makes the brain slower and less nimble), and whether exercise increases the levels of noggin (which improves the brain's ability). For each of the results quoted in parts (a), (b), and (c), interpret the information about the p-value in terms of evidence for the effect. (a) "Exercise improved Y-maze performance in most mice by the 7 th day of exposure, with further increases after 10 days for all mice tested \((p<.01)\) (b) "After only two days of running, BMP ... was reduced \(\ldots\) and it remained decreased for all subsequent time-points \((p<.01)\)." (c) "Levels of noggin ... did not change until 4 days, but had increased 1.5 -fold by \(7-10\) days of exercise \((p<.001)\)." (d) Which of the tests appears to show the strongest statistical effect? (e) What (if anything) can we conclude about the effects of exercise on mice?

Give null and alternative hypotheses for a population proportion, as well as sample results. Use StatKey or other technology to generate a randomization distribution and calculate a p-value. StatKey tip: Use "Test for a Single Proportion" and then "Edit Data" to enter the sample information. Hypotheses: \(H_{0}: p=0.5\) vs \(H_{a}: p \neq 0.5\) Sample data: \(\hat{p}=42 / 100=0.42\) with \(n=100\)

Suppose you want to find out if reading speed is any different between a print book and an e-book. (a) Clearly describe how you might set up an experiment to test this. Give details. (b) Why is a hypothesis test valuable here? What additional information does a hypothesis test give us beyond the descriptive statistics we discuss in Chapter \(2 ?\) (c) Why is a confidence interval valuable here? What additional information does a confidence interval give us beyond the descriptive statistics of Chapter 2 and the results of a hypothesis test described in part (b)? (d) A similar study \(^{53}\) has been conducted, and reports that "the difference between Kindle and the book was significant at the \(p<.01\) level, and the difference between the iPad and the book was marginally significant at \(p=.06 . "\) The report also stated that "the iPad measured at \(6.2 \%\) slower reading speed than the printed book, whereas the Kindle measured at \(10.7 \%\) slower than print. However, the difference between the two devices [iPad and Kindle] was not statistically significant because of the data's fairly high variability." Can you tell from the first quotation which method of reading (print or e-book) was faster in the sample or do you need the second quotation for that? Explain the results in your own words.

The same sample statistic is used to test a hypothesis, using different sample sizes. In each case, use StatKey or other technology to find the p-value and indicate whether the results are significant at a \(5 \%\) level. Which sample size provides the strongest evidence for the alternative hypothesis? Testing \(H_{0}: p_{1}=p_{2}\) vs \(H_{a}: p_{1}>p_{2}\) using \(\hat{p}_{1}-\hat{p}_{2}=0.45-0.30=0.15\) with each of the following sample sizes: (a) \(\hat{p}_{1}=9 / 20=0.45\) and \(\hat{p}_{2}=6 / 20=0.30\) (b) \(\hat{p}_{1}=90 / 200=0.45\) and \(\hat{p}_{2}=60 / 200=0.30\) (c) \(\hat{p}_{1}=900 / 2000=0.45\) and \(\hat{p}_{2}=600 / 2000=0.30\)
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