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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).

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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.

The 2012 general Social Survey asked a large number of people how much time they spent watching TV each day. The mean number of hours was 3.09 with a standard deviation of 2.76. Assume that in a sample of 30 teenagers, the sample standard deviation of daily TV time is 3.7 hours, and that the population of TV watching times is normally distributed. Can you conclude that the population standard deviation of TV watching times for teenagers differs from 2.76? Use α = 0.10 level of significance.

Perform a hypothesis test to determine if the population standard deviation of TV watching times for teenagers differs from 2.76 hours. This example demonstrates the chi-square test for variance with a sample size of 30 teenagers and a significance level of 0.10.

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