Question

Hours Spent Watching Television According to Nielsen Media Research, children (ages 2-11) spend an average of 21 hours 30 minutes watching television per week while teens (ages 12-17) spend an average of 20 hours 40 minutes. Based on the sample statistics shown, is there sufficient evidence to conclude a difference in average television watching times between the two groups? Use \(\alpha=0.01 .\) $$ \begin{array}{lcc}{} & {\text { Children }} & {\text { Teens }} \\ \hline \text { Sample mean } & {22.45} & {18.50} \\ {\text { Sample variance }} & {16.4} & {18.2} \\ {\text { Sample size }} & {15} & {15}\end{array} $$

Short answer

No, there is not enough evidence to conclude a difference in average TV watching times at \(\alpha = 0.01\).

Step-by-step solution

State the Hypotheses

First, we need to establish the null hypothesis and the alternative hypothesis. The null hypothesis \(H_0\) states that there is no difference in the average television watching times between children and teens: \(H_0: \mu_1 = \mu_2\). The alternative hypothesis \(H_a\) states that there is a difference: \(H_a: \mu_1 eq \mu_2\).

Calculate the Test Statistic

We'll use the two-sample t-test formula for comparing the means of two independent groups. The formula for the t-statistic is: \[t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}\]Where \(\bar{x}_1 = 22.45\), \(\bar{x}_2 = 18.50\), \(s_1^2 = 16.4\), \(s_2^2 = 18.2\), \(n_1 = 15\), \(n_2 = 15\).

Perform the Calculation

Substitute the values into the formula:\[t = \frac{22.45 - 18.50}{\sqrt{\frac{16.4}{15} + \frac{18.2}{15}}}\]Calculate the differences and the square root:\[t = \frac{3.95}{\sqrt{1.0933 + 1.2133}} = \frac{3.95}{1.471}\]\[t \approx 2.69\]

Determine the Critical Value and Compare

Given \(\alpha = 0.01\) and a two-tailed test, with \(n_1 + n_2 - 2 = 28\) degrees of freedom, we find the critical t-value using a t-distribution table. The critical value for \(\alpha = 0.01\) is approximately \(\pm 2.763\). Since our calculated \(t\) statistic (2.69) is less than 2.763, we do not reject the null hypothesis.

Conclusion

Since the calculated t-statistic is less than the critical value, there is not enough evidence to reject the null hypothesis. Therefore, we conclude that there is insufficient evidence to suggest a significant difference in average television watching times between children and teens at \(\alpha = 0.01\).

Key concepts

Two-Sample t-Test

The two-sample t-test is a powerful statistical method used when we want to compare the means of two independent groups. In our example, it helps us determine if the average time spent watching television per week by children is significantly different from that of teens.

Three key assumptions in this test include:

• The populations should have normal distributions.
• The variances of the two populations are equal.
• The samples are independent.

In practice, the two-sample t-test involves calculating a t-statistic, which tells us how far apart our sample means are in terms of standard errors. This is why the formula includes terms that account for sample variability and size. A larger absolute t-statistic indicates a bigger difference between the means.

Critical Value

The critical value in hypothesis testing is a cutoff point that determines the threshold for deciding whether to reject or not reject the null hypothesis. In the given exercise, the significance level, denoted by \( \alpha \), is set at 0.01, indicating a 1% risk of making a Type I error (rejecting a true null hypothesis).

The critical value is the boundary that separates extreme values from those that are considered normal. Since we are conducting a two-tailed test, we must check the t-distribution table to find the critical value for a 0.01 significance level and the appropriate degrees of freedom. Values beyond this critical point suggest significant evidence against the null hypothesis, prompting its rejection.

t-Distribution

The t-distribution is a type of probability distribution that is symmetric and bell-shaped, like the normal distribution, but with thicker tails. It is especially useful when dealing with small sample sizes, as it provides a more accurate estimate of variability. The shape of the t-distribution depends on the degrees of freedom, which we'll discuss next.

This distribution becomes very useful when conducting a t-test because it allows us to assess the probability of obtaining a t-statistic as extreme as what we've calculated, under the assumption that the null hypothesis is true. As the sample size increases, the t-distribution becomes closer to a normal distribution.

Degrees of Freedom

Degrees of freedom (df) are a critical concept in statistics that refer to the number of values in a calculation that are free to vary. In the context of a two-sample t-test, the degrees of freedom help determine the shape of the t-distribution used in hypothesis testing.

For independent sample t-tests, df is calculated as the sum of the sample sizes minus two, which accounts for the estimation of two group means—one for each group. For our exercise, with both groups having 15 samples each, the degrees of freedom are calculated as \(df = n_1 + n_2 - 2 = 28\). This calculation impacts the critical value we use to determine significance, affecting whether the difference between sample means can be considered statistically significant.