Use a chi-squared test to test the claim about the population variance or standard deviation at the given level of significance and using the given sample statistics.
Claim: ơ2 ≤ 60; α = 0.025. Sample statistics: s2 = 72.7, n = 15
You might try this formula for your chi-square test:
Chi-square = (n - 1)(sample variance)/(value specified in the null hypothesis)
n = 15
sample variance = 72.7
value specified in the null hypothesis = 60
Chi-square = (15 - 1)(72.7)/60
Finish the calculation for your chi-square test statistic.
Degrees of freedom is equal to n - 1, which is 14.
Check a chi-square distribution table using alpha = 0.025, with 14 degrees of freedom to determine your critical or cutoff value.
If the test statistic does not exceed the critical or cutoff value from the table, then the null is not rejected. If the test statistic exceeds the critical or cutoff value from the table, then the null is rejected.
I hope this will help.
To determine if the claim about the population variance (σ^2) or standard deviation (σ) is supported or not, we can use a chi-squared test.
The null hypothesis (H0) is that the population variance/standard deviation is less than or equal to 60 (σ^2 ≤ 60). The alternative hypothesis (Ha) is that the population variance/standard deviation is greater than 60 (σ^2 > 60).
To perform the chi-squared test, we need to calculate the test statistic and compare it to the critical value.
1. Calculate the test statistic (chi-squared statistic):
- The test statistic for a chi-squared test of variance is calculated as: χ^2 = (n - 1) * s^2 / σ^2, where n is the sample size, s^2 is the sample variance, and σ^2 is the hypothesized population variance.
- In this case, the sample statistics given are s^2 = 72.7 and n = 15. The hypothesized population variance is σ^2 = 60.
- Plugging in the values, we can calculate the test statistic as: χ^2 = (15 - 1) * 72.7 / 60 = 18.44.
2. Determine the critical value:
- The critical value for a chi-squared test is obtained from the chi-squared distribution table, with degrees of freedom (df) equal to n - 1.
- In this case, the degrees of freedom are df = 15 - 1 = 14.
- Since the level of significance (α) is given as 0.025, we need to find the critical value that corresponds to an area of 0.025 in the right tail of the chi-squared distribution with 14 degrees of freedom.
- Using the chi-squared distribution table or a statistical software, we find that the critical value is approximately 26.12.
3. Compare the test statistic to the critical value:
- If the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
- In this case, the test statistic is χ^2 = 18.44, and the critical value is 26.12.
- Since 18.44 is less than 26.12, we fail to reject the null hypothesis.
4. Conclusion:
- Based on the results, there is not enough evidence to support the claim that the population variance/standard deviation is greater than 60 (σ^2 > 60) at the α = 0.025 level of significance.
- Therefore, we can conclude that the available data does not provide sufficient evidence to suggest that the population variance exceeds 60.