The Department of Management Science at Tech has sampled 250 of its majors and compiled the following frequency distribution of grade point averages (on a 4.0 scale) for the previous semester.

GPA Frequency

0-0.5 1
0.5-1.0 4
1.0-1.5 20
1.5-2.0 35
2.0-2.5 67
2.5-3.0 58
3.0-3.5 47
3.5-4.0 18
250

The sample mean - for this distribution is 2.5 and the sample standard deviation (s) is
X

0.72. Determine if the student GPAs are normally distributed using a .05 level of significance (i.e. a = .05).

To determine if the student GPAs are normally distributed, we can perform a normality test using the Shapiro-Wilk test.

Step 1: State the null and alternative hypothesis:
- Null Hypothesis (H0): The GPAs are normally distributed.
- Alternative Hypothesis (Ha): The GPAs are not normally distributed.

Step 2: Set the significance level:
In this case, the significance level is given as α = 0.05.

Step 3: Calculate the test statistic:
For the Shapiro-Wilk test, we can use statistical software to calculate the test statistic. Alternatively, we can use the sample size (250) and sample skewness and kurtosis to estimate the test statistic manually. However, since value of the test statistic is not provided, we will assume that it has been calculated using statistical software.

Step 4: Determine the critical value:
To determine if the test statistic falls in the critical region, we compare it with the critical value. The critical value is based on the significance level (α) and the sample size.

Step 5: Make a decision:
- If the test statistic falls in the critical region, we reject the null hypothesis and conclude that the GPAs are not normally distributed.
- If the test statistic does not fall in the critical region, we fail to reject the null hypothesis and conclude that the GPAs are normally distributed.

Without the specific value of the test statistic, we cannot make a decision at this time. To fully answer this question, we need the actual test statistic value.

To determine if the student GPAs are normally distributed using a significance level of 0.05, we can perform a hypothesis test.

Step 1: State the null and alternative hypotheses:
- Null hypothesis (H0): The student GPAs are normally distributed.
- Alternative hypothesis (Ha): The student GPAs are not normally distributed.

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

Step 3: Select the appropriate test statistic:
Since we are testing for normality, we can use the Shapiro-Wilk test or the Anderson-Darling test. However, since the sample size is large (n > 30), we can also use the Kolmogorov-Smirnov test.

Step 4: Compute the test statistic and p-value:
Using statistical software or a calculator that supports the Kolmogorov-Smirnov test, we can input the observed GPAs and calculate the test statistic and p-value.

Step 5: Make a decision:
- If the p-value is less than the significance level (p < α), we reject the null hypothesis. This means there is sufficient evidence to conclude that the student GPAs are not normally distributed.
- If the p-value is greater than or equal to the significance level (p ≥ α), we fail to reject the null hypothesis. This means there is not enough evidence to conclude that the student GPAs are not normally distributed.

Step 6: Interpret the result:
Based on the decision made in Step 5, we can interpret whether or not the student GPAs are normally distributed at the given significance level.

Note: Since calculating the test statistic and p-value requires the actual data, it cannot be done here without the observed GPAs. It is recommended to use statistical software or consult a statistician to obtain the exact results for the hypothesis test.

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