Question

Each person in a random sample of 228 male teenagers and a random sample of 306 female teenagers was asked how many hours he or she spent online in a typical week (Ipsos, January 25,2006 ). The sample mean and standard deviation were \(15.1\) hours and \(11.4\) hours for males and \(14.1\) and \(11.8\) for females. a. The standard deviation for each of the samples is large, indicating a lot of variability in the responses to the question. Explain why it is not reasonable to think that the distribution of responses would be approximately normal for either the population of male teenagers or the population of female teenagers. Hint: The number of hours spent online in a typical week cannot be negative. b. Given your response to Part (a), would it be appropriate to use the two- sample \(t\) test to test the null hypothesis that there is no difference in the mean number of hours spent online in a typical week for male teenagers and female teenagers? Explain why or why not. c. If appropriate, carry out a test to determine if there is convincing evidence that the mean number of hours spent online in a typical week is greater for male teenagers than for female teenagers. Use a \(.05\) significance level.

Short answer

The large variability and the range of possible responses makes it unlikely that the distribution of responses is approximately normal for either male or female teens. Despite this, the two-sample t test may still be used due to the large sample sizes. Based on the t test performed with the given data, a conclusion should be drawn about whether there's significant evidence to support the hypothesis that male teenagers spend more time online in a typical week than female teenagers, with the decision based on whether the p-value is less than the significance level of .05.

Step-by-step solution

Understanding the distribution of responses

The distribution of responses (i.e., the number of hours spent online) for either the male or female population won't be approximately normal. This is because the values can't be less than zero, so the distribution is not symmetric around the mean. The variability is large, indicating that there are many data points far from the mean, which would make the distribution skewed to the right.

Considering the use a two-sample t test

Despite the concern about the non-normal distribution of responses, a two-sample t test may be appropriate because of the Central Limit Theorem, which states that the distribution of sample means will tend to be normal even if the distribution of the individual data points is not, as long as the sample size is large enough. In this case, the sample size for each group (228 for males and 306 for females) may be considered large enough to use the t test, as a rule of thumb often mentioned is that if sample sizes are greater than 30, the t test can be used.

Performing the two-sample t test

The null hypothesis for the two-sample t test is that the two populations have the same mean. Here, this translates to there being no difference in the average time spent online by male and female teens. The alternative hypothesis is that males spend significantly more time online. We calculate the t statistic using the formula: \(t = \frac{{(mean1 - mean2) - (hypothesized difference)}}{{\sqrt{{(std dev1^2 / n1) + (std dev2^2 / n2)}}}}\), where the values correspond to male (1) and female (2) data. Plug in the given values to calculate the t value and then use a t-table or software to find the p-value for this t-statistic with appropriate degrees of freedom (n1 + n2 - 2). Compare the p-value to the given significance level (.05) to decide whether to reject the null hypothesis.