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Practice Exam: MS 5217 Statistical Data Analysis

1 Short Question (50 points)

1. If P(A ∪ B) = 0.5 and P(A ∩ B) = 0.5, then P(A) = P(B).

2. Let X and Y be independent random variables. Then the variance of the sum is given by V ar(X + Y) = V ar(X) + V ar(Y).

3. The expectation of X minus 2Y is just the expectation of X minus twice the expectation of Y , that is E(X − 2Y) = E(X) − 2E(Y).

4. Suppose that there’s a 5% chance that it snows tomorrow and a 80% chance that the Chicago bears play their football game tomorrow given that it snows. The probability that they play tomorrow is then 80%.

5. If X has a normal distribution with mean 3 and standard deviation 5, then Z = has a standard normal distribution.

6. As the sample size (nunber of data observations) increases, the confifidence interval of sample mean narrows, holding all else the same.

7. A semi-conductor company knows from experience that 0.2% of chips will have imperfec-tions. Suppose it makes 1000 such chips, then the probability that at least one is imperfect is over 95%.

8. If we find a sample of data with exactly 95% observations between µ ± 1.96 × σ, we can say this sample follows a normal distribution.

9. A friend claims she can tell the difference between Evian and Dasani bottled water. Suppose p is the probability she can identify Evian correctly. In a random experiment with 100 repeated tests, the proportion that she can correctly identified the Evian water is p = 0.6. Then you can reject the null hypothesis that p =  at the 95% level.

10. The p-value is the probability that the Null hypothesis is true.

2 Long Question (40 points)

1. A screening test for high blood pressure, corresponding to a diastolic blood pressure of 90mm Hg or higher, produced the following probability table

(a) What’s the probability that a random person has hypertension?

(b) What’s the probability that someone tests positive on the test?

(c) Given a person who tests positive, what is the probability that they have hypertension?

(d) What would happen to your probability of having hypertension given you tested positive if you initially thought you had a 50% chance of having hypertension.

2. The Chicago Cubs are having a great season. So far they’ve won 72 out of the 100 games played so far. You also have the expert opinion of Bob the sports analysis. He tells you that he thinks the Cubs will win. Historically his predictions have a 60% chance of coming true.

(a) Calculate the probability that the Cubs will win given Bob’s prediction

(b) Suppose you now learn that it’s a home game and that 60% of the wins of the Cubs are played at home. What’s you updated probability that the Cubs will win their game?