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Chapter 10: Problem 71

An article titled "Teen Boys Forget Whatever It Was" appeared in the Australian newspaper The Mercury (April 21, 1997). It described a study of academic performance and attention span and reported that the mean time to distraction for teenage boys working on an independent task was 4 minutes. Although the sample size was not given in the article, suppose that this mean was based on a random sample of 50 teenage Australian boys and that the sample standard deviation was \(1.4\) minutes. Is there convincing evidence that the average attention span for teenage boys is less than 5 minutes? Test the relevant hypotheses using \(\alpha=.01\).

Short Answer

Expert verified
Without the actual calculations, a definitive short answer cannot be provided. The decision to reject or fail to reject the null hypothesis would be based on the comparison between the calculated t-value and the critical t-value for the one-tailed t-test at the .01 significance level.

Step by step solution

01

State the hypotheses

The null hypothesis \(H_0\) is that the population mean \(\mu = 5\). This means that it is assumed that the average attention span for teenage boys is 5 minutes. The alternative hypothesis \(H_1\) is the claim that we are testing for which states that \(\mu < 5\). That is the average attention span is less than 5 minutes.
02

Compute the test statistic

The test statistic is a t-score (t). This is calculated using the formula: \[ t = \frac{{\bar{x} - \mu}}{{s / \sqrt{n}}}\]Where \(\bar{x}\) = sample mean = 4 minutes, \(\mu\) = hypothesized population mean = 5 minutes, \(s\) = sample standard deviation = 1.4 minutes, and \(n\) = sample size = 50.Plug the numbers into the formula to get the t-score.
03

Determine the critical value

The significance level, \(\alpha\), is given as .01. Given that this is a one-tailed test with 49 degrees of freedom (df = n - 1), the critical value of t can be found in the t-distribution table or by using statistical software. This is the value that the calculated t-score must exceed for the result to be statistically significant.
04

Make a decision

Compare the calculated t-score from Step 2 to the critical value from Step 3. If the t-score is more extreme in the negative direction (since this is a test for less than) than the critical value, the null hypothesis is rejected in favor of the alternative. This would provide convincing evidence that the average attention span for teenage boys is indeed less than 5 minutes. If not, the conclusion is to not reject the null hypothesis.

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Most popular questions from this chapter

Assuming a random sample from a large population, for which of the following null hypotheses and sample sizes \(n\) is the large-sample \(z\) test appropriate: a. \(H_{0}: p=.2, n=25\) b. \(H_{0}: p=.6, n=210\) c. \(H_{0}: p=.9, n=100\) d. \(H_{0}: p=.05, n=75\)

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In a national survey of 2013 adults, 1590 responded that lack of respect and courtesy in American society is a serious problem, and 1283 indicated that they believe that rudeness is a more serious problem than in past years (Associated Press, April 3, 2002 ). Is there convincing evidence that less than three-quarters of U.S. adults believe that rudeness is a worsening problem? Test the relevant hypotheses using a significance level of \(.05\).
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