Question

As in Assignments 1 through 3, consider the following set of depression scores from a sample of 25 freshmen at the beginning of the semester (the assessment tool uses a 0-10 ratio scale, with 0 indicating no detectable level of depression; assume that the underlying population is normally distributed): 2, 4, 8, 6, 7, 1, 3, 3, 9, 5, 2, 5, 4, 7, 1, 6, 6, 3, 2, 8, 4, 3, 5, 4, 10. In a previous assignment, you computed this sample’s mean, x̄, as 4.72. 1. As the newly hired director of the University Counseling Center, you believe that more resources are necessary to address students’ mental health: in your opinion, the overall depression level among freshmen is high, based on the sample of 25 first-year students (x-bar = 4.72). However, the Dean of Students provides you with data from the entire freshmen population that indicates a comparatively lower level of depression; specifically, her data indicates that the average is µ = 3.8 on the above-mentioned depression inventory. a. Assuming that the standard deviation in the freshmen population is σ = 3, test the viability of the Dean of Students’ statement against your own belief. That is, test the null hypothesis that µ = 3.8 against an appropriate directional alternative hypothesis. Set the level of significance to 0.05. Step 1: State the null and alternative hypothesis H0: µ = 3.8 Ha: µ > 3.8 Step 2: Set the criteria for a decision: α = 0.05 critical value: 1.645 Step 3: Compute the test statistic Test statistic: z = (x̄ - µ) / (σ / √n) Step 4: Make a decision AND interpret the result b. Now, re-assess the viability of the Dean of Students’ statement by using the sample statistics you computed in previous assignments: x̄ = 4.72 and s = 2.48. That is, determine an appropriate test statistic and critical value and, if called for, revise your conclusion. critical value: 1.711 Test statistic: t = (x̄ - µ) / (s / √n) Conclusion (if different from Part a): c. For this context, explain the meaning of a Type I and a Type II error (be specific) and then make an argument for which one, Type I or Type II, is the more severe and consequential error in this situation.

As in Assignments 1 through 3, consider the following set of depression scores from a sample of 25 freshmen at the beginning of the semester (the assessment tool uses a 0-10 ratio scale, with 0 indicating no detectable level of depression; assume that the underlying population is normally distributed): 2, 4, 8, 6, 7, 1, 3, 3, 9, 5, 2, 5, 4, 7, 1, 6, 6, 3, 2, 8, 4, 3, 5, 4, 10.

In a previous assignment, you computed this sample’s mean, x̄, as 4.72.

1. As the newly hired director of the University Counseling Center, you believe that more resources are necessary to address students’ mental health: in your opinion, the overall depression level among freshmen is high, based on the sample of 25 first-year students (x-bar = 4.72). However, the Dean of Students provides you with data from the entire freshmen population that indicates a comparatively lower level of depression; specifically, her data indicates that the average is µ = 3.8 on the above-mentioned depression inventory.

a. Assuming that the standard deviation in the freshmen population is σ = 3, test the viability of the Dean of Students’ statement against your own belief. That is, test the null hypothesis that µ = 3.8 against an appropriate directional alternative hypothesis. Set the level of significance to 0.05.

Step 1: State the null and alternative hypothesis
H0: µ = 3.8
Ha: µ > 3.8

Step 2: Set the criteria for a decision:
α = 0.05
critical value: 1.645

Step 3: Compute the test statistic
Test statistic: z = (x̄ - µ) / (σ / √n)

Step 4: Make a decision AND interpret the result

b. Now, re-assess the viability of the Dean of Students’ statement by using the sample statistics you computed in previous assignments: x̄ = 4.72 and s = 2.48. That is, determine an appropriate test statistic and critical value and, if called for, revise your conclusion.

critical value: 1.711
Test statistic: t = (x̄ - µ) / (s / √n)

Conclusion (if different from Part a):

c. For this context, explain the meaning of a Type I and a Type II error (be specific) and then make an argument for which one, Type I or Type II, is the more severe and consequential error in this situation.

Added by Jason G.

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