The grades in the final exam of a random sample of 25 students produce a standard deviation, s = 9.5.
a) Construct a 90% confidence interval estimate for the population variance, .
b) Can we infer at 10% level of significance that the population variance is significantly greater than 100?
To construct a confidence interval for the population variance, we can use the chi-square distribution. The formula for the confidence interval is given by:
( n - 1 ) s^2 / chi-squared-upper , ( n - 1 )s^2 / chi-squared-lower
Where:
- n is the sample size (in this case, 25)
- s is the sample standard deviation (in this case, 9.5)
- chi-squared-upper is the chi-square value corresponding to the upper tail probability (in this case, 0.05, since the confidence level is 90%)
- chi-squared-lower is the chi-square value corresponding to the lower tail probability (in this case, 0.05)
a) To construct a 90% confidence interval estimate for the population variance, we need to find the chi-square values for the upper and lower tail probabilities. Using a chi-square table or a statistical software, we can find that the chi-square value for a upper tail probability of 0.05 with 24 degrees of freedom is approximately 38.081 and the chi-square value for a lower tail probability of 0.05 is approximately 10.644.
Plugging in the values, we get:
(24 * (9.5)^2) / 38.081, (24 * (9.5)^2) / 10.644
Simplifying further: (2286 / 38.081), (2286 / 10.644)
The confidence interval for the population variance is approximately (60.12, 214.90).
b) To infer whether the population variance is significantly greater than 100 at a 10% level of significance, we can conduct a hypothesis test. The hypotheses are as follows:
Null hypothesis (H0): Population variance <= 100
Alternative hypothesis (Ha): Population variance > 100
To conduct this test, we can use the chi-square distribution. We calculate the test statistic value using the following formula:
chi-square = (n - 1) * s^2 / sigma
Where:
- n is the sample size (in this case, 25)
- s is the sample standard deviation (in this case, 9.5)
- sigma is the hypothesized population variance (in this case, 100)
Plugging in the values, we get:
chi-square = (24 * (9.5)^2) / 100
Simplifying further: 216.96 / 100 = 2.17
The test statistic value is 2.17.
Next, we need to find the critical chi-square value for a 10% level of significance with 24 degrees of freedom. Using a chi-square table or a statistical software, we find that the critical chi-square value is approximately 36.415.
Since the test statistic value (2.17) is less than the critical chi-square value (36.415), we do not have enough evidence to reject the null hypothesis. Therefore, we cannot infer at a 10% level of significance that the population variance is significantly greater than 100.