### COMMENTS ON SECOND EXAM    (M 358K, Sp 06 Smith)

Grades were lower than on the first exam. There are three likely causes:
1. The material is harder.
2. The time span between the two exams was long.
3. The exam was a little long -- some people were rushed at the end.
To take (3) into account, I have set grade cut-offs 5 points lower than usual:

A: 85 - 100    (5)
B: 75 - 84     (8)
C: 65 - 74    (5)
D: 55- 64    (3)
F: < 55        (9)

Top quarter: ≥ 77
Median:        71.5
Lower quarter: ≤ 52

Please Note: My office hours for Tuesday, May 4 are cancelled. I will have substitute office hours for this class only on that day from 11:30 to 1.

EXAMPLES OF GRADING ON PROBLEM 1 OF EXAM 2

Part a: (7 points) Note: This was graded fairly leniently. Student got 5 points if they seemed to understand the basic idea but missed some details.

7 points: The probability, assuming μ = 2, of obtaining a test statistic at least as large as the one obtained from our sample, when considering all possible simple random samples of the same size from the same population, is 0.08.

6 points: The P-value is the probability, found as 0.08, of getting a test statistics as extreme as the one we observed, given μ = 2.
(Student did not clarify what the probability space was.)

5 points: The probability, computed assuming μ = 2, that the test statistic will have a value at least as extreme as that actually observed.
(Student did not clarify what the probability space was and did not translate “at least as extreme” into the context given.)

4 points: 0.08 is the probability that we reject the null hypothesis when in fact μ = 2. The P-value is the probability that the outcome is as extreme or at least as extreme as our observed outcome, assuming μ = 2.
(Second sentence was similar to student who got 5 points, but this student also had an extra sentence confusing P-value with significance level.)

3 points: The P-value is the probability (in this case, 0.08) of obtaining a test statistics (where a test statistics is calculated from a SRS of size n) that is equal to or greater than the one from this sample. It is the probability that μ = 2.
(Student gets off to a pretty good start in the first sentence, but then shows a serious misunderstanding in the second sentence.)

2 points: There is a probability of 0.08 of obtaining the same value of z or larger if we choose another sample.
(Student indicates awareness of repeated sampling, but does not indicate that the probability assumes μ = 2.)

0 points: The P-value is the largest probability for which we would accept the null hypothesis μ = 2.
(Student has confused P-value and significance level, and also misunderstood significance level.)

0 points: The p-value is the probability of getting μ = 2 with data that is given.
(Student seems to have serious misunderstanding.)

0 points: The p-value is the probability that will reject the null hypothesis.
(Student has confused P-value and significance level, and also misunderstood significance level.)

0 points: A P-value is a number that is used to either support or reject H0.
(This says what one uses the P-value for, but doesn’t say what the P-value is.)

Part b: (8 points)

8 points: The interval (2.20, 2.80) was produced by a method which, for 99% of all possible simple random samples of the same size from the population of UT students who have earned between 60 and 90 hours, will give an interval containing the mean of the GPA’s of all UT students who have earned between 60 and 90 hours.

7 points: We’ve used a procedure that for 99% o fall possible simple random sample of size n from the population will produce a CI that contains the population mean. IN this case, we are 99% confident that the true mean GPA of UT students who have earned between 60 and 90 credit hours lies between 2.20 and 2.80.
(Student doesn’t clarify the population in the first sentence, and then in the second sentence just uses alternate terminology but doesn’t show clearly that they understand.)

5 points: A confidence interval can be described in the following way: In repeated sampling of SRS’ containing the true mean GPA, I will produce an interval containing the true mean GPA 99% of the time. Also, in this specific case, I can say that I am 99% confident that the true mean lies in the interval above.
(Student neglects to specific the population that samples must be taken from, but more importantly, does not indicate that there is a procedure involved. Also, the last sentence just introduces alternate terminology that may obscure more than clarify.)

4 points: A method was used so that, for 99% of all possible SRS’s of UT students who have earned between 60 and 90 credit hours, an interval is given that contains the true mean GPA. That interval is (2.20, 2.80).
(Student gets off to a good start in first sentence, although does not specify that samples must be of the same size. But from the second sentence, it is unclear whether or not the student is saying tha the interval will always be the one given, or whether the interval given is just the one produced from one sample.)

0 points: This confidence interval tells us that we can be 99% confident that the population mean GPA f all UT students who have earned between 60 and 90 credit hours falls within our range of (2.20, 2.80).
(Student has just used alternate terminology, but not explained the concept.)

0 points: When repeating the experiment many times using sample of the same size, the actual mean GPA will fall in the above interval 99% of the time.
(Student does not make the very important point that the confidence interval varies from sample to sample.)

0 points: If the experiment were repeated many times, you would expect 99% of your sample means to fall into this interval.
(Student  shows serious misunderstandings – one similar to the above student, and an additional one that sample means rather than the population mean are what we are trying to capture.)