Exam Grades. 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 á = 0.05, can it be concluded that the variance in grades exceeds 100?

92.3
96.7
88.5
89.4
69.5
79.2
76.9
72.8
72.9
65.2
67.5
68.7
49.1
52.8
75.8

To determine whether the variance in grades exceeds 100, we can use a chi-square test for variance.

Step 1: State the hypotheses.
- Null hypothesis (H₀): The variance in grades is equal to or less than 100 (σ² ≤ 100).
- Alternative hypothesis (H₁): The variance in grades exceeds 100 (σ² > 100).

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

Step 3: Calculate the test statistic.
We can use the chi-square formula for variance: chi-square = (n-1) * sample variance / population variance.

Sample grades:
92.3, 96.7, 88.5, 89.4, 69.5, 79.2, 76.9, 72.8, 72.9, 65.2, 67.5, 68.7, 49.1, 52.8, 75.8

First, calculate the sample variance:
Step 3.1: Calculate the mean of the sample.
Mean = (92.3 + 96.7 + 88.5 + 89.4 + 69.5 + 79.2 + 76.9 + 72.8 + 72.9 + 65.2 + 67.5 + 68.7 + 49.1 + 52.8 + 75.8) / 15
= 979.3 / 15
= 65.2867

Step 3.2: Calculate the sum of squared differences from the mean.
Sum of squared differences = (92.3 - 65.2867)² + (96.7 - 65.2867)² + (88.5 - 65.2867)² + ... + (75.8 - 65.2867)²

Step 3.3: Calculate the sample variance.
Sample variance = Sum of squared differences / (n-1)

Using the sample, calculate the sample variance.

Step 3.4: Calculate the test statistic.
chi-square = (n-1) * sample variance / population variance

Where n is the sample size.

Step 4: Determine the critical value.
The critical value is obtained from the chi-square distribution table with (n-1) degrees of freedom and a significance level of 0.05.

Step 5: Compare the test statistic with 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.

Based on the information provided, we can follow the steps above to determine if the variance in grades exceeds 100. Note that the values used in the calculation will need to be provided to complete the calculations accurately.

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

Step 1: Define the null and alternative hypotheses.
- Null hypothesis (H0): The variance in grades is less than or equal to 100.
- Alternative hypothesis (H1): The variance in grades exceeds 100.

Step 2: Determine the test statistic.
Since we are dealing with sample variances, we will use the chi-squared test statistic. The chi-squared test statistic for testing the variance is calculated as:

Chi-squared = (n - 1) * sample variance / population variance

where n is the sample size.

Step 3: Set the significance level.
The significance level (á) is given in the problem as 0.05. This is the probability of rejecting the null hypothesis when it is true.

Step 4: Find the critical value.
The critical value for a chi-squared test with a given significance level and degrees of freedom can be found using a chi-squared distribution table or calculator. Degrees of freedom are equal to the sample size minus one.

Step 5: Calculate the test statistic.
Here, we don't have the population variance, so we have to estimate it using the sample variance. The sample variance (s^2) can be calculated by finding the sum of the squared differences between each grade and the mean, and then dividing it by the sample size minus one.

Step 6: Compare the test statistic with 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.

Let's calculate the test statistic and compare it with the critical value to determine if the variance in grades exceeds 100.

Given mid-term grades:
92.3, 96.7, 88.5, 89.4, 69.5, 79.2, 76.9, 72.8, 72.9, 65.2, 67.5, 68.7, 49.1, 52.8, 75.8

Step 5: Calculate the sample variance.
First, find the sample mean:
mean = (92.3 + 96.7 + 88.5 + 89.4 + 69.5 + 79.2 + 76.9 + 72.8 + 72.9 + 65.2 + 67.5 + 68.7 + 49.1 + 52.8 + 75.8) / 15 = 74.91

Next, calculate the sum of squared differences from the mean:
SS = (92.3 - 74.91)^2 + (96.7 - 74.91)^2 + (88.5 - 74.91)^2 + ... + (75.8 - 74.91)^2 = 1551.0774

Finally, calculate the sample variance:
sample variance = SS / (n - 1) = 1551.0774 / (15 - 1) = 103.40516

Step 2: Calculate the test statistic.
test statistic = (n - 1) * sample variance / population variance = 14 * 103.40516 / 100 = 14.466504

Step 4: Find the critical value.
The critical value for a chi-squared test with 14 degrees of freedom and a significance level of 0.05 is 23.6848.

Step 6: Compare the test statistic with the critical value.
Since the test statistic (14.466504) is less than the critical value (23.6848), we fail to reject the null hypothesis.

Thus, at a significance level of 0.05, we cannot conclude that the variance in grades exceeds 100.

The midterm test scores for the seventh-period typing class are listed below.

85 77 93 91 74 65 68 97 88 59 74 83 85 72 63 79