Question

Suppose Ted, a sociologist, is interested in comparing the amount of time that husbands and wives in the United States each spend on social media. He suspects that the average time spent on social media by husbands differs from the average time spent by wives. Ted randomly selects 41 heterosexual married couples from a U.S. census database and asks each spouse how much time they spend on social media in a typical week. Next, he computes the difference in social media usage time between the husband and wife for each couple by subtracting the wife's time online from her husband's time online. Ted's set of calculated differences has a sample mean of -1.22 h and a sample standard deviation of 1.17 h. Although the distribution of intramarital social media usage differences in the population is unknown, the sample data have a single peak and are fairly symmetric. The normal probability plot and boxplot show a slight skew to the left with no strong outliers. Select the statement that accurately evaluates whether or not a matched-pairs t-test is valid in Ted's experiment.

1. A matched-pairs t-test is valid, despite the sample being a small representation of the population, because the sample is a simple random sample and has a distribution with a single peak.
2. A matched-pairs t-test is valid because a simple random sample of sufficient size is obtained from a large population and no outliers are in the set of differences, which have an unknown population standard deviation.
3. A matched-pairs t-test is not valid because information about the data distributions and outlier status of the separate husband and wife samples is not provided.
4. A matched-pairs t-test is not valid because the sampling distribution is skewed left, which makes the distribution less symmetric, and, therefore, the assumption of normality has been violated.
5. A matched-pairs t-test is not valid because this experiment needs a two-sample t-test using the social media usage data collected from the sample of husbands and from the sample of wives.

Answer 1

1. A matched-pairs t-test is valid, despite the sample being a small representation of the population, because the sample is a simple random sample and has a distribution with a single peak.

Explanation:

The matched-pairs test is valid, for the reasons given in choice 1. Here's why:

• We do have matched pairs, not a 2-sample t-test, because each two are paired by the house they live in. Husband and wife live together: it's safe to pair them. (This rules out option 5.)

• Check conditions: The sample is large enough (fulfilling the sample size condition). The sample data is fairly normal, although we don't know the population data, and the sample size is over 40, so we consider it a fairly large sample (fulfilling the nearly normal condition). We don't know about outliers, but we'll have to assume Ted doesn't have any, because they aren't mentioned.