x is normally distributed and that sample of 20 yields a mean of 42 and a variance of 25 test the hypothesis that the population standard deviation is 8 at 5% significance level.
To test the hypothesis that the population standard deviation is 8 at a 5% significance level, we can use a chi-square test.
Step 1: State the hypotheses:
- Null hypothesis (H0): The population standard deviation is 8.
- Alternative hypothesis (H1): The population standard deviation is not 8.
Step 2: Set the significance level (alpha):
In this case, the significance level is 5% or 0.05.
Step 3: Determine the test statistic:
Since we are testing the population standard deviation, we use the chi-square distribution. The test statistic follows the chi-square distribution with n-1 degrees of freedom, where n is the sample size.
Given that the sample size is 20, the degrees of freedom will be 20 - 1 = 19.
Step 4: Calculate the test statistic:
The test statistic can be calculated using the formula:
chi-square = (n - 1) * sample variance / population variance
In this case, n = 20, sample variance = 25, and population variance (hypothesized value) = 8^2 = 64.
chi-square = (20 - 1) * 25 / 64
chi-square = 475 / 64
chi-square ≈ 7.42 (rounded to two decimal places)
Step 5: Determine the critical value:
To determine the critical value from the chi-square distribution table, we need to find the value that corresponds to the significance level (alpha) and the degrees of freedom (19).
At a 5% significance level, the critical value is approximately 30.14 (found in the chi-square distribution table).
Step 6: Make a decision:
If the test statistic (7.42) is less than the critical value (30.14), we fail to reject the null hypothesis (H0) and conclude that there is not enough evidence to suggest that the population standard deviation is different from 8.
If the test statistic is greater than or equal to the critical value, we reject the null hypothesis (H0) and conclude that there is enough evidence to suggest that the population standard deviation is different from 8.
In this case, the test statistic (7.42) is less than the critical value (30.14), so we fail to reject the null hypothesis.
Therefore, at a 5% significance level, there is not enough evidence to conclude that the population standard deviation is different from 8.