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Chapter 8: Problem 27

A taste test is done to see whether a person can tell Coke from Pepsi. In each case, 20 random and independent trials are done (half with Pepsi and half with Coke) in which the person determines whether she or he is drinking Coke or Pepsi. One person gets 13 right out of 20 trials. Which of the following is the correct figure to test the hypothesis that the person can tell the difference? Explain your choice.

Short Answer

Expert verified
The correct figure required to test the hypothesis that the person can tell the difference is the 'Z' value or the test statistic from step 2. It quantifies the difference between the success rate obtained from the experiments against the success rate under the null hypothesis. By comparing this with the critical value, one can decide whether to accept or reject the null hypothesis.

Step by step solution

01

- Null and Alternative Hypothesis

Firstly, define the null hypothesis (H0) and the alternative hypothesis (H1). The definition can be as follows: \(H_0\): The person is guessing (success rate is .5), \(H_1\): The person is not guessing (success rate is not .5)
02

- Calculate Observed Test Statistic

Next, calculate the observed test statistic. The formula for observed test statistic when testing proportions is \(Z = (\widehat{p} - P_0) / \sqrt{ P_0 * (1 - P_0) / n}\), where \(\widehat{p}\) = x/n, n = number of trials, x = number of successful trials and \(P_0\) is the success probability under the null hypothesis. Substituting the given values we can calculate Z.
03

- Critical Region and Conclusion

Now find the critical value for the 5% significance level (typical value used if not specified) usually denoted as Z_0.05. If the calculated Z falls in the critical region (greater than Z_0.05 or less than -Z_0.05 for a two-tailed test), then we reject the null hypothesis in favor of the alternative. If not, there is not sufficient evidence to suggest the person can accurately distinguish between the drinks.

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Most popular questions from this chapter

A teacher giving a true/false test wants to make sure her students do better than they would if they were simply guessing, so she forms a hypothesis to test this. Her null hypothesis is that a student will get \(50 \%\) of the questions on the exam correct. The alternative hypothesis is that the student is not guessing and should get more than \(50 \%\) in the long run. $$\begin{aligned}&\mathrm{H}_{0}: p=0.50 \\\&\mathrm{H}_{\mathrm{a}}: p>0.50\end{aligned}$$ A student gets 30 out of 50 questions, or \(60 \%\), correct. The p-value is \(0.079\). Explain the meaning of the \(\mathrm{p}\) -value in the context of this question.

A community college used enrollment records of all students and reported that that the percentage of the student population identifying as female in 2010 was \(54 \%\) whereas the proportion identifying as female in 2018 was \(52 \%\). Would it be appropriate to use this information for a hypothesis test to determine if the proportion of students identifying as female at this college had declined? Explain.

Sir William Blackstone (1723-1780) wrote influential books on common law. He made this statement: "All presumptive evidence of felony should be admitted cautiously; for the law holds it better that ten guilty persons escape, than that one innocent party suffer." Keep in mind that the null hypothesis in criminal trials is that the defendant is not guilty. State which of these errors (in blue) is the first type of error (rejecting the null hypothesis when it is actually true) and which is the second type of error.

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

By establishing a small value for the significance level, are we guarding against the first type of error (rejecting the null hypothesis when it is true) or guarding against the second type of error?
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