Based on the image, you're working on a hypothesis test related to the standard deviation of TV watching times for teenagers. The task asks for the null hypothesis ($H_0$) and the alternative hypothesis ($H_1$) regarding the population standard deviation. Here's a breakdown of how to proceed:

Problem Information:

• Population standard deviation ($\sigma$) from previous studies: $2.76$ hours.
• Sample size: $30$ teenagers.
• Sample standard deviation ($s$): $3.7$ hours.
• Significance level: $\alpha = 0.10$.
• The goal is to determine if the standard deviation of TV watching times for teenagers differs from $2.76$.

Hypotheses:

• Null hypothesis $H_0$: The population standard deviation is equal to 2.76 hours. This can be written as: $H_0: \sigma = 2.76$

• Alternative hypothesis $H_1$: The population standard deviation differs from 2.76 hours (since the question asks if it differs, it suggests a two-tailed test). This can be written as: $H_1: \sigma \neq 2.76$

Type of Test:

This is a two-tailed test because we are checking for a difference in either direction (whether the standard deviation is larger or smaller than 2.76).

Once you set these hypotheses, the next steps would involve calculating the test statistic (likely using a chi-square test for variance) and comparing it to the critical values or p-value for the significance level $\alpha = 0.10$.

Would you like help with the next parts of the question, like calculating the test statistic or interpreting the results?

Relative Questions:

1. What is the formula for the chi-square test statistic in hypothesis testing for variance?
2. How do you find the critical values for a chi-square distribution?
3. How does sample size affect the hypothesis test for variance?
4. What are the assumptions for performing a chi-square test on variance?
5. How does the significance level impact the decision in a hypothesis test?

Tip:

For hypothesis tests about standard deviations or variances, always check if the sample is normally distributed because the chi-square test relies on that assumption.