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Hypothesis Testing: Does Chance explain the Results?

Hypothesis Tests

An experimenter suspects that a certain die is “loaded;” that is, the chances that the die lands on different faces are not all equal. Recall that dice are made with the sum of the numbers of spots on opposite sides equal to 7: 1 and 6 are opposite each other, 2 and 5 are opposite each other, and 3 and 4 are opposite each other.

The experimenter decides to test the null hypothesis that the die is fair against the alternative hypothesis that it is not fair, using the following test. The die will be rolled 50 times, independently. If the die lands with one spot showing 13 times or more, or 3 times or fewer, the null hypothesis will be rejected.

Under the null hypothesis, the distribution of the number of times the die lands showing one spot is binomial with parameters n=50, p=1/6 

Under the null hypothesis, the expected number of times the die lands showing one spot is 8.3333 and the standard error of the number of times the die lands showing one spot is (Q6) 

Problem 4

The significance level of this test is (Q7) 

Problem 5

The power of this test against the alternative hypothesis that the chance the die lands with one spot showing is 29.28%, the chance the die lands with six spots showing is 4.05%, and the chances the die lands with two, three, four, or five spots showing each equal 1/6, is (Q8) 

Problem 6

The power of this test against the alternative hypothesis that the chance the die lands with two spots showing is 17.42%, the chance the die lands with five spots showing is 15.91%, and the chances the die lands with one, three, four, or six spots showing each equal 1/6, is (Q9) 

Suppose that to have power against a wider variety of alternatives, the experimenter decides to base the test on the maximum number of times any face shows, instead of just the number of times one spot shows. That is, she will roll the die 50 times and calculate

(number of times die lands showing one spot)
(number of times die lands showing two spots)
(number of times die lands showing three spots)
(number of times die lands showing four spots)
(number of times die lands showing five spots)
and (number of times die lands showing six spots).

She will reject the null hypothesis if the largest (maximum) of those 6 random numbers is greater than 17.

Problem 7

Under the null hypothesis, the distribution of the maximum number of times any face shows is (Q10)   ? 

A: geometric 

C: normal 

D: binomial 

E: negative binomial 

F: none of the above  

A manufacturer of computer memory chips produces chips in lots of 1000. If nothing has gone wrong in the manufacturing process, at most 7 chips each lot would be defective, but if something does go wrong, there could be far more defective chips. If something goes wrong with a given lot, they discard the entire lot. It would be prohibitively expensive to test every chip in every lot, so they want to make the decision of whether or not to discard a given lot on the basis of the number of defective chips in a simple random sample. They decide they can afford to test 100 chips from each lot. You are hired as their statistician.

There is a tradeoff between the cost of eroneously discarding a good lot, and the cost of warranty claims if a bad lot is sold. The next few problems refer to this scenario.

Problem 8

(Continues previous problem.) A type I error occurs if a good lot is discarded 

Problem 9

(Continues previous problem.) A type II error occurs if a bad lot is not discarded 

Problem 10

(Continues previous problem.) Under the null hypothesis, the number of defective chips in a simple random sample of size 100 has a (Q14)   ? 

A: hypergeometric 

B: geometric 

C: binomial 

E: negative binomial 

F: none of the above  

distribution, with parameters (Q15)   ? 

A: p=7/1000 

B: mean=7/1000 , SD=833.727 

C: N=1,000, G<=7, n=100 

E: p=7/1000, r=100 

F: none of the above 

Problem 11

(Continues previous problem.) To have a chance of at most 3% of discarding a lot given that the lot is good, the test should reject if the number of defectives in the sample of size 100 is greater than or equal to (Q16) 

Problem 12

(Continues previous problem.) In that case, the chance of rejecting the lot if it really has 30 defective chips is (Q17) 

Problem 13

(Continues previous problem.) In the long run, the fraction of lots with 7 defectives that will get discarded erroneously by this test is (Q18) 

Problem 14

(Continues previous problem.) The smallest number of defectives in the lot for which this test has at least a 97% chance of correctly detecting that the lot was bad is (Q19) 

(Continues previous problem.) Suppose that whether or not a lot is good is random, that the long-run fraction of lots that are good is 93%, and that whether each lot is good is independent of whether any other lot or lots are good. Assume that the sample drawn from a lot is independent of whether the lot is good or bad. To simplify the problem even more, assume that good lots contain exactly 7 defective chips, and that bad lots contain exactly 30 defective chips.

Problem 15

(Continues previous problem.) The number of lots the manufacturer has to produce to get one good lot that is not rejected by the test has a (Q20)   ? 

A: geometric 

B: binomial 

C: hypergeometric 

D: normal 

F: none of the above  

distribution, with parameters (Q21)   ? 

A: p=96.89% 

B: N=1,000, G=(7+30)/2, n=100 

C: n=100, p=90.66% 

D: p=90.66%, r=100 

F: p=97.48% 

G: mean=(7+30)/2, SD=4.3 

H: n=100, p=96.89% 

I: p=90.66%

J: p=97.48%, r=100 

K: n=100, p=97.48% 

L: none of the above 

Problem 16

(Continues previous problem.) The expected number of lots the manufacturer must make to get one good lot that is not rejected by the test is (Q22) 

Problem 17

(Continues previous problem.) With this test and this mix of good and bad lots, among the lots that pass the test, the long-run fraction of lots that are actually bad is (Q23) 

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