TVaholicsAccording to the American Time Use Survey, the typical American spends 154.8 minutes (2.58 hours) per day watching television. A survey of 50 Internet users results in a mean time watching television per day of 128.7 minutes, with a standard deviation of 46.5 minutes. Conduct the appropriate test to determine if Internet users spend less time watching television at the α = 0.05 level of significance.


Source: Norman H. Nie and D. Sunshine Hillygus. “Where Does Internet Time Come From? A Reconnaissance.” IT & Society, 1(2).

Verified step by step guidance

1. Identify the null hypothesis (H_0) and the alternative hypothesis (H_a). Since we want to test if Internet users spend less time watching television, set H_0: \( \mu = 154.8 \) minutes and H_a: \( \mu < 154.8 \) minutes, where \( \mu \) is the true mean time Internet users spend watching TV.
2. Determine the significance level \( \alpha = 0.05 \) and note that this is a one-tailed (left-tailed) test because the alternative hypothesis is looking for a decrease.
3. Calculate the test statistic using the formula for a one-sample t-test since the population standard deviation is unknown and the sample size is 50 (which is moderately large): \[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \] where \( \bar{x} = 128.7 \) minutes (sample mean), \( \mu_0 = 154.8 \) minutes (population mean under H_0), \( s = 46.5 \) minutes (sample standard deviation), and \( n = 50 \) (sample size).
4. Find the critical t-value from the t-distribution table with \( df = n - 1 = 49 \) degrees of freedom at the \( \alpha = 0.05 \) significance level for a one-tailed test.
5. Compare the calculated t-statistic to the critical t-value: - If the t-statistic is less than the critical t-value, reject the null hypothesis, indicating that Internet users spend significantly less time watching television. - Otherwise, fail to reject the null hypothesis.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. In this question, the null hypothesis states that Internet users spend the same or more time watching TV as the general population, while the alternative suggests they spend less. The test evaluates sample data against a significance level (α = 0.05) to make this decision.

One-Sample t-Test

A one-sample t-test compares the sample mean to a known population mean when the population standard deviation is unknown. Here, it tests if the mean TV watching time of Internet users (sample) is significantly less than the general population mean of 154.8 minutes. The test uses the sample mean, sample standard deviation, and sample size to calculate a t-statistic.

Significance Level and p-Value

The significance level (α) is the threshold for rejecting the null hypothesis, commonly set at 0.05. The p-value measures the probability of observing the sample data, or something more extreme, if the null hypothesis is true. If the p-value is less than α, the null hypothesis is rejected, indicating a statistically significant difference.