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Chapter 4: Problem 74

A study \(^{20}\) conducted in June 2015 examines ownership of tablet computers by US adults. A random sample of 959 people were surveyed, and we are told that 197 of the 455 men own a tablet and 235 of the 504 women own a tablet. We want to test whether the survey results provide evidence of a difference in the proportion owning a tablet between men and women. (a) State the null and alternative hypotheses, and define the parameters. (b) Give the notation and value of the sample statistic. In the sample, which group has higher tablet ownership: men or women? (c) Use StatKey or other technology to find the pvalue.

Short Answer

Expert verified
The null hypothesis is \(P_M = P_W\) and the alternative hypothesis is \(P_M \neq P_W\). The difference in proportions is 0.033, showing a higher tablet ownership in women. The p-value to test the hypotheses must be calculated using statistical software.

Step by step solution

01

State the Null and Alternative Hypotheses and Define the Parameters

The null hypothesis (\(H_0\)) is that there is no difference in the proportions of tablet ownership between men and women. This is symbolized as \(P_M = P_W\), where \(P_M\) is the proportion of men who own a tablet and \(P_W\) is the proportion of women who own a tablet. The alternative hypothesis (\(H_A\)) is that there is a difference in the proportions of tablet ownership between men and women. This is symbolized as \(P_M \neq P_W\). The parameters in this case are \(P_M\) and \(P_W\).
02

Define the Sample Statistic and Identify the Group with Higher Tablet Ownership

The sample statistic is the difference in proportions between tablet owners who are men and those who are women. This is calculated as follows: Ratio of men owning tablets = 197 / 455 = 0.433. Ratio of women owning tablets = 235 / 504 = 0.466. Difference in proportions = 0.466 - 0.433 = 0.033. The group with higher tablet ownership in the sample is Women.
03

Calculation of p-value

To find the p-value, you need to conduct a test of proportions. Using appropriate statistical software or methods like the Z-test, you input the number of owners and the sample sizes for both groups. You then conduct a two-proportion Z-test or the Chi-Square test for independence (depending on the software). The value generated by this test will be your p-value, indicating the probability that the observed difference could have come about by chance under the null hypothesis. Please note that to carry out this step it is necessary to use statistical software: the exact way to enter values will depend on this software, and the specific number can't be given without running this test.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null and Alternative Hypotheses
Understanding the null and alternative hypotheses is crucial when conducting a difference in proportions test, like in the textbook exercise concerning tablet ownership by gender. The null hypothesis (\(H_0\)) serves as a starting point, implying no effect or no difference, which would mean both genders own tablets at the same rate (\(P_M = P_W\)).

The alternative hypothesis (\(H_A\)), on the other hand, suggests that there is a difference in the proportions (\(P_M eq P_W\)). This hypothesis attempts to demonstrate what we suspect might be true, that tablet ownership varies between men (\(P_M\)) and women (\(P_W\)). The objective of the test is to determine if we have enough evidence to reject the null hypothesis and accept the alternative hypothesis.
Sample Statistic
A sample statistic is a numerical summary about a sample that often estimates the unknown parameter of interest in the population. In the context of the tablet ownership study, the sample statistic is the difference in the proportion of tablet ownership between the sample of men and women.

This difference is calculated by subtracting the proportion of male tablet owners (\(197 / 455 = 0.433\)) from the proportion of female tablet owners (\(235 / 504 = 0.466\)), which gives a difference of 0.033. This difference indicates which group, in the sample, has higher ownership, and in this case, it is women, as the sample statistic is positive.
P-value Calculation
The p-value is a crucial component in hypothesis testing, representing the probability that the sample statistic would be as extreme as it is, or more so, if the null hypothesis were true. To calculate the p-value in our tablet ownership study, one would typically use a Z-test for two proportions, due to the normal distribution of sample proportions.

The calculation involves determining how many standard deviations away the observed sample statistic is from the expected value under the null hypothesis. Although the specific p-value is not provided due to the lack of statistical software output, it would be the key to deciding whether the difference in tablet ownership by gender observed in the sample could be due to random chance.
Statistical Significance
Statistical significance plays an important role in determining whether the findings from a study can be considered valid. If the calculated p-value is less than a pre-determined threshold (commonly \(\alpha = 0.05\)), the result is considered statistically significant.

This means the observed difference is unlikely to have occurred by chance, and we have enough evidence to reject the null hypothesis. In the context of the tablet ownership exercise, if the p-value were below this threshold, it would suggest a statistically significant difference in tablet ownership between men and women.
Tablet Ownership by Gender
The study seeks to determine whether there's an actual disparity between male and female tablet ownership. Initial results from the sample indicate a higher proportion of women owning tablets.

This particular exercise demonstrates the process of examining observed data to determine if it reflects a true trend in the broader population or occurs merely by chance. By applying tests for the difference in proportions, we understand better how gender may or may not influence technology ownership patterns, an insight that contributes to market research and sociological studies.

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Most popular questions from this chapter

Indicate whether it is best assessed by using a confidence interval or a hypothesis test or whether statistical inference is not relevant to answer it. (a) What percent of US voters support instituting a national kindergarten through \(12^{\text {th }}\) grade math curriculum? (b) Do basketball players hit a higher proportion of free throws when they are playing at home than when they are playing away? (c) Do a majority of adults riding a bicycle wear a helmet? (d) On average, were the 23 players on the 2014 Canadian Olympic hockey team older than the 23 players on the 2014 US Olympic hockey team?

For each situation described, indicate whether it makes more sense to use a relatively large significance level (such as \(\alpha=0.10\) ) or a relatively small significance level (such as \(\alpha=0.01\) ). Testing to see whether taking a vitamin supplement each day has significant health benefits. There are no (known) harmful side effects of the supplement.

The Ignorance Surveys were conducted in 2013 using random sampling methods in four different countries under the leadership of Hans Rosling, a Swedish statistician and international health advocate. The survey questions were designed to assess the ignorance of the public to global population trends. The survey was not just designed to measure ignorance (no information), but if preconceived notions can lead to more wrong answers than would be expected by random guessing. One question asked, "In the last 20 years the proportion of the world population living in extreme poverty has \(\ldots, "\) and three choices were provided: 1) "almost doubled" 2) "remained more or less the same," and 3) "almost halved." Of 1005 US respondents, just \(5 \%\) gave the correct answer: "almost halved." 34 We would like to test if the percent of correct choices is significantly different than what would be expected if the participants were just randomly guessing between the three choices. (a) What are the null and alternative hypotheses? (b) Using StatKey or other technology, construct a randomization distribution and compute the p-value. (c) State the conclusion in context.

Influencing Voters When getting voters to support a candidate in an election, is there a difference between a recorded phone call from the candidate or a flyer about the candidate sent through the mail? A sample of 500 voters is randomly divided into two groups of 250 each, with one group getting the phone call and one group getting the flyer. The voters are then contacted to see if they plan to vote for the candidate in question. We wish to see if there is evidence that the proportions of support are different between the two methods of campaigning. (a) Define the relevant parameter(s) and state the null and alternative hypotheses. (b) Possible sample results are shown in Table 4.3 . Compute the two sample proportions: \(\hat{p}_{c},\) the proportion of voters getting the phone call who say they will vote for the candidate, and \(\hat{p}_{f},\) the proportion of voters getting the flyer who say they will vote for the candidate. Is there a difference in the sample proportions? (c) A different set of possible sample results are shown in Table 4.4. Compute the same two sample proportions for this table. (d) Which of the two samples seems to offer stronger evidence of a difference in effectiveness between the two campaign methods? Explain your reasoning. $$ \begin{array}{lcc} \hline & \begin{array}{c} \text { Will Vote } \\ \text { Sample A } \end{array} & \text { for Candidate } & \begin{array}{l} \text { Will Not Vote } \\ \text { for Candidate } \end{array} \\ \hline \text { Phone call } & 152 & 98 \\ \text { Flyer } & 145 & 105 \\ \hline \end{array} $$ $$ \begin{array}{lcc} \text { Sample B } & \begin{array}{c} \text { Will Vote } \\ \text { for Candidate } \end{array} & \begin{array}{c} \text { Will Not Vote } \\ \text { for Candidate } \end{array} \\ \hline \text { Phone call } & 188 & 62 \\ \text { Flyer } & 120 & 130 \\ \hline \end{array} $$

Does consuming beer attract mosquitoes? Exercise 4.17 on page 268 discusses an experiment done in Africa testing possible ways to reduce the spread of malaria by mosquitoes. In the experiment, 43 volunteers were randomly assigned to consume either a liter of beer or a liter of water, and the attractiveness to mosquitoes of each volunteer was measured. The experiment was designed to test whether beer consumption increases mosquito attraction. The report \(^{30}\) states that "Beer consumption, as opposed to water consumption, significantly increased the activation \(\ldots\) of \(A n\). gambiae [mosquitoes] ... \((P<0.001)\)." (a) Is this convincing evidence that consuming beer is associated with higher mosquito attraction? Why or why not? (b) How strong is the evidence for the result? Explain. (c) Based on these results, it is reasonable to conclude that consuming beer causes an increase in mosquito attraction? Why or why not?
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