Question

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 min. 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 \(1.4\) min. Is there convincing evidence that the average attention span for teenage boys is less than 5 min? Test the relevant hypotheses using \(\alpha=.01\).

Short answer

The short answer will depend on the calculated p-value. If p-value < .01, then the average attention span for teenage boys is less than 5 minutes. Otherwise, there is not enough convincing evidence that the average attention span for teenage boys is less than 5 minutes.

Step-by-step solution

Formulate the Hypotheses

The null hypothesis \(H_0\) is the statement that the average attention span for teenage boys equals 5 minutes. Mathematically, this can be written as \(H_0: \mu = 5\). The alternative hypothesis \(H_A\) is the statement that the average attention span of teenage boys is less than 5 minutes (H_A: \mu < 5).

Perform the t-test

The test statistic for a one-sample t-test is: \(t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}\) where \(\bar{x}\) is the sample mean, \(\mu_0\) is the hypothesized population mean, \(s\) is the sample standard deviation, and \(n\) is the sample size. Substituting, we have: \(t = \frac{4 - 5}{1.4 / \sqrt{50}}\)

Compute the p-value

Find the p-value associated with the observed value of the test statistic (t) using a statistics software or t-distribution table.

Make a decision

If the p-value is less than the significance level (0.01), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

Interpret the result

If the null hypothesis is rejected, it suggests that there is convincing evidence that the average attention span for teenage boys is less than 5 minutes. If the null hypothesis is not rejected, it suggests that there is not enough evidence to conclude that the average attention span for teenage boys is less than 5 minutes.