To determine whether the population standard deviation of TV watching times for teenagers differs from 2.76 hours, we can perform a chi-square test for the population standard deviation. The test will compare the sample standard deviation (3.1 hours) with the hypothesized population standard deviation (2.76 hours).
### Hypotheses:
- Null Hypothesis (\(H_0\)): \(\sigma = 2.76\) (The population standard deviation is 2.76 hours)
- Alternative Hypothesis (\(H_a\)): \(\sigma \neq 2.76\) (The population standard deviation is not 2.76 hours)
### Test Statistic:
The chi-square test statistic for a sample variance is given by:
\[ \chi^2 = \frac{(n-1)s^2}{\sigma_0^2} \]
where:
- \(n\) is the sample size,
- \(s\) is the sample standard deviation,
- \(\sigma_0\) is the hypothesized population standard deviation.
Given the data:
- Sample size (\(n\)) = 45
- Sample standard deviation (\(s\)) = 3.1 hours
- Hypothesized population standard deviation (\(\sigma_0\)) = 2.76 hours
Plugging in these values:
\[ \chi^2 = \frac{(45-1) \cdot (3.1)^2}{(2.76)^2} \]
\[ \chi^2 = \frac{44 \cdot 9.61}{7.6176} \]
\[ \chi^2 = \frac{422.84}{7.6176} \]
\[ \chi^2 \approx 55.51 \]
### Degrees of Freedom:
The degrees of freedom (\(df\)) for the chi-square test is given by \(n - 1\). Thus,
\[ df = 45 - 1 = 44 \]
### Critical Values:
Since the significance level (\(\alpha\)) is 0.10, we will use the chi-square distribution to find the critical values for a two-tailed test. We need to find \(\chi^2_{\alpha/2, df}\) and \(\chi^2_{1-\alpha/2, df}\).
Using a chi-square table or a calculator for \(\alpha = 0.10\) and \(df = 44\):
- The lower critical value (\(\chi^2_{0.05, 44}\)): approximately 30.804
- The upper critical value (\(\chi^2_{0.95, 44}\)): approximately 59.341
### Decision Rule:
- If \(\chi^2\) lies outside the range \([30.804, 59.341]\), we reject \(H_0\).
- If \(\chi^2\) lies within the range \([30.804, 59.341]\), we fail to reject \(H_0\).
### Conclusion:
Our test statistic \(\chi^2 \approx 55.51\) lies within the range \([30.804, 59.341]\).
Therefore, at the \(\alpha = 0.10\) level of significance, we do not reject the null hypothesis. There is not enough evidence to conclude that the population standard deviation of TV watching times for teenagers differs from 2.76 hours.
The 2012 General Social Survey asked a large number of people how many times 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 45 teenagers, the sample standard deviation of the daily TV time is 3.1 hours, and that the population of TV-watching times is normally distributed. Can we conclude that the population standard deviation of TV watching times for teenagers differs from 2.76. Use the a= 0.10 level of significance. Find the critical values and state them
3 answers
Find the critical values for this two tailed test
Sure, let's accurately identify the critical values for the chi-square distribution given an \(\alpha = 0.10\) level of significance and 44 degrees of freedom.
For a two-tailed test with \(\alpha = 0.10\), we divide the significance level across both tails of the distribution. This means:
- The lower critical region comprises \(\alpha/2 = 0.10/2 = 0.05\) to the left tail.
- The upper critical region comprises \(\alpha/2 = 0.10/2 = 0.05\) to the right tail.
To find the critical values, we will use a chi-square table or calculator designed to provide critical values for the chi-square distribution.
1. **Lower critical value** (\(\chi^2_{0.05, 44}\)):
- This is the chi-square value corresponding to the 0.05 percentile (lower 5%) with 44 degrees of freedom.
- From the chi-square table or calculator, \(\chi^2_{0.05, 44} \approx 30.796\).
2. **Upper critical value** (\(\chi^2_{0.95, 44}\)):
- This is the chi-square value corresponding to the 95th percentile (upper 5%) with 44 degrees of freedom.
- From the chi-square table or calculator, \(\chi^2_{0.95, 44} \approx 59.342\).
### Summary of Critical Values:
- Lower critical value: 30.796
- Upper critical value: 59.342
These critical values define the range for our decision rule:
- If the calculated chi-square test statistic is less than 30.796 or greater than 59.342, we reject the null hypothesis \(H_0\).
- If the calculated chi-square test statistic is between 30.796 and 59.342, we fail to reject the null hypothesis \(H_0\).
Thus, at the \(\alpha = 0.10\) level of significance with 44 degrees of freedom, the critical values for this two-tailed test are 30.796 and 59.342.
For a two-tailed test with \(\alpha = 0.10\), we divide the significance level across both tails of the distribution. This means:
- The lower critical region comprises \(\alpha/2 = 0.10/2 = 0.05\) to the left tail.
- The upper critical region comprises \(\alpha/2 = 0.10/2 = 0.05\) to the right tail.
To find the critical values, we will use a chi-square table or calculator designed to provide critical values for the chi-square distribution.
1. **Lower critical value** (\(\chi^2_{0.05, 44}\)):
- This is the chi-square value corresponding to the 0.05 percentile (lower 5%) with 44 degrees of freedom.
- From the chi-square table or calculator, \(\chi^2_{0.05, 44} \approx 30.796\).
2. **Upper critical value** (\(\chi^2_{0.95, 44}\)):
- This is the chi-square value corresponding to the 95th percentile (upper 5%) with 44 degrees of freedom.
- From the chi-square table or calculator, \(\chi^2_{0.95, 44} \approx 59.342\).
### Summary of Critical Values:
- Lower critical value: 30.796
- Upper critical value: 59.342
These critical values define the range for our decision rule:
- If the calculated chi-square test statistic is less than 30.796 or greater than 59.342, we reject the null hypothesis \(H_0\).
- If the calculated chi-square test statistic is between 30.796 and 59.342, we fail to reject the null hypothesis \(H_0\).
Thus, at the \(\alpha = 0.10\) level of significance with 44 degrees of freedom, the critical values for this two-tailed test are 30.796 and 59.342.
3 answers