Question

Your statistics professor assigns nice tests, cruel tests or diabolical tests. Nice tests are the easiest, diabolical tests are the hardest, and cruel tests are in between. On cruel tests, scores are normally distributed with μc= 52 and σc= 14. Nice and diabolical tests have different μ and different σ. Your class has 49 students. The average score for the class of 49 students is Xbar.

(a) If the first test is cruel, what is the probability that X bar is between 54 ~ 58?

(b) The average score on the first test is 47. Test the null hypothesis that the first test was cruel. Use α = .05 and a two-sided test to test this null hypothesis.

(c) Suppose that nice tests also have scores that are normally distributed, but they have a different μ and σ, with μn = 55 and σn = 10.5. For the second test of the semester, you plan to do another hypothesis test, again with the null hypothesis that the test is cruel. If the second test is in fact nice, not cruel, what is the probability you will make a type 2 error in your hypothesis test?

(d) The professor says that the final test will have a 50% chance of being cruel and a 50% chance of being nice. For this test, the professor does not report the class average, instead he randomly picks our one exam and tells you the score. If the randomly chosen exam is above an 80, what is the probability it was nice?

Answer 1

(a)

The z-score for is

The z-score for is

The probability that X bar is between 54 and 58 is

(b)

(c)

(d)

If the test is cruel then the probability that score is above 80 is:

The z-score for C = 80 is

So,

P(Score above 80 | cruel) = P(z > 2) = 0.0228

If the test is nice then the probability that score is above 80 is:

The z-score for N = 80 is

So,

P(Score above 80 | nice) = P(z > 2.38) = 0.0087

And we have

P(cruel) = P(nice) = 0.50

By the Baye's theorem we have

P(nice | score above 80) = [ P(Score above 80 | nice)P(nice) ] / [ P(Score above 80 | nice)P(nice) + [ P(Score above 80 | cruel)P(cruel) ] ] = [ 0.0087 *0.5 ] / [ 0.0087 *0.5 + 0.0228 * 0.5 ] = 0.2762