A statistics professor is used to having a variance in his class grades of no more than 100. He feels that his current group of students is different, and so he examines a random sample of midterm grades (listed below). At 0.05 alpha level can it be conculded that the variance in the grades differs from 100? Write a 90% confidence level of the variance using the information given

92.3 89.4 76.9 65.2. 49.1. 96 7. 69.5. 72.8. 77.5 52.8. 8.85. 79.2. 72.9. 68.7. 75.8

To determine whether the variance in the grades differs from 100, we need to conduct a hypothesis test and calculate a confidence interval. Here's how you can do it:

Step 1: Hypothesis Testing
1. Set up the null and alternative hypotheses:
- Null Hypothesis (H0): Variance in grades is equal to 100.
- Alternative Hypothesis (Ha): Variance in grades is not equal to 100.

2. Select an appropriate statistical test. In this case, we use a Chi-square test since we are comparing variances.

3. Calculate the test statistic:
- Sample Variance (s^2) = sum((x - mean)^2) / (n - 1), where x represents the individual grades in the sample and n is the sample size.
- Test Statistic (chi-square) = (n - 1) * s^2 / variance

4. Determine the critical value for the test statistic. Since the problem states an alpha level of 0.05, which corresponds to a 95% confidence level, we will use a significance level of 0.05 (alpha = 0.05). Look up the critical value in the chi-square distribution table with degrees of freedom (df = n-1).

5. Compare the test statistic with the critical value. If the test statistic is greater than the critical value or falls outside the chosen rejection region, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

Step 2: Confidence Interval
1. Calculate the confidence interval for the population variance using the chi-square distribution:
- Lower bound: ((n - 1) * s^2) / Chi-square upper critical value (1 - alpha/2)
- Upper bound: ((n - 1) * s^2) / Chi-square lower critical value (alpha/2)

2. Convert the variance to a standard deviation, if needed:
- Standard Deviation = sqrt(variance)

So now, let's calculate the necessary values using the provided information:

Given grades: 92.3 89.4 76.9 65.2 49.1 96 7 69.5 72.8 77.5 52.8 8.85 79.2 72.9 68.7 75.8
Sample size (n) = 16
Hypothesized variance (assumed population variance) = 100

Step 1: Hypothesis Testing
Calculate the test statistic:
Sample Variance (s^2) = sum((x - mean)^2) / (n - 1)
mean = sum(x) / n
Test Statistic (chi-square) = (n - 1) * s^2 / variance

Step 2: Confidence Interval
Calculate the lower and upper bounds of the confidence interval.

I hope this helps you get started on solving the problem! Let me know if you need further assistance with the calculations.

To determine if the variance in the grades differs from 100, we will need to perform a hypothesis test.

Step 1: State the null and alternative hypotheses.
The null hypothesis (H0): The variance in the grades is equal to 100.
The alternative hypothesis (Ha): The variance in the grades differs from 100.

Step 2: Set the significance level (alpha).
The given alpha level is 0.05.

Step 3: Calculate the test statistic.
To calculate the test statistic, we will use the chi-square distribution. The formula to calculate the test statistic is as follows:

chi-square = (n-1) * s^2 / sigma^2

where n is the sample size, s^2 is the sample variance, and sigma^2 is the hypothesized population variance.

Given:
Sample size (n) = 15
Sample data: 92.3, 89.4, 76.9, 65.2, 49.1, 96.7, 69.5, 72.8, 77.5, 52.8, 8.85, 79.2, 72.9, 68.7, 75.8
Hypothesized population variance (sigma^2) = 100

Using these values, we can calculate the test statistic.

Step 4: Determine the critical value.
Since the alpha level is 0.05 and we have a two-tailed test, we need to find the critical values that correspond to the alpha/2 level. We can look up the critical values in the chi-square distribution table with (n-1) degrees of freedom.

For a 90% confidence level, the critical values for a chi-square distribution with 14 degrees of freedom are 6.5706 (upper tail) and 23.6848 (lower tail).

Step 5: Make a decision.
If the calculated test statistic falls within the critical value range, we fail to reject the null hypothesis. If it falls outside the range, we reject the null hypothesis.

Step 6: Calculate the confidence interval.
To calculate the confidence interval for the variance, we can use the formula:

CI = [(n-1) * s^2 / chi-square upper, (n-1) degrees of freedom, alpha/2, (n-1) * s^2 / chi-square lower, (n-1) degrees of freedom, 1 - alpha/2]

Using the calculated test statistic, the critical values, and the sample variance, we can calculate the confidence interval for the variance.

Now let's perform the calculations.

Step 3 (continued): Calculate the test statistic.
n = 15
s^2 = (92.3^2 + 89.4^2 + 76.9^2 + 65.2^2 + 49.1^2 + 96.7^2 + 69.5^2 + 72.8^2 + 77.5^2 + 52.8^2 + 8.85^2 + 79.2^2 + 72.9^2 + 68.7^2 + 75.8^2) / 15 = 1818.3153
sigma^2 = 100

chi-square = (15-1) * 1818.3153 / 100 = 324.9702

Step 4 (continued): Determine the critical value.
The critical values are:
Upper tail critical value: 6.5706
Lower tail critical value: 23.6848

Step 5 (continued): Make a decision.
Since the calculated test statistic (324.9702) is greater than the upper tail critical value (6.5706), we reject the null hypothesis. There is sufficient evidence to conclude that the variance in the grades differs from 100 at the 0.05 alpha level.

Step 6 (continued): Calculate the confidence interval.
CI = [(15-1) * 1818.3153 / 6.5706, (15-1) * 1818.3153 / 23.6848] = [6563.5444, 18254.1349]

Therefore, with 90% confidence, the variance in the grades is estimated to be between 6563.5444 and 18254.1349.