To test the sociologist's claim that people watch a mean of 3 hours of TV per day, we will perform a hypothesis test using the following null and alternative hypotheses:
- Null hypothesis \((H_0)\): \(\mu = 3\) (The mean number of hours spent watching TV per day is equal to 3 hours)
- Alternative hypothesis \((H_a)\): \(\mu < 3\) (The mean number of hours spent watching TV per day is less than 3 hours)
Given:
- Sample size \(n = 1314\)
- Sample mean \(\bar{x} = 2.85\)
- Sample standard deviation \(s = 2.62\)
- Population mean under the null hypothesis \(\mu_0 = 3\)
We will use a one-sample t-test, and the test statistic can be calculated using the formula:
\[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \]
Substituting the values:
\[ t = \frac{2.85 - 3}{2.62 / \sqrt{1314}} \]
Calculating the standard error:
\[ s / \sqrt{n} = \frac{2.62}{\sqrt{1314}} \approx \frac{2.62}{36.244} \approx 0.0723 \]
Now substitute back into the test statistic formula:
\[ t = \frac{2.85 - 3}{0.0723} \approx \frac{-0.15}{0.0723} \approx -2.07 \]
Thus, the value of the test statistic is approximately:
\[ t \approx -2.07 \]
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