Hypothesis Testing: K-S test & Test for Normality

1. A manufacturer of power meters, which are used to regulate energy thresholds of a data-communications system, claims that when its production process is operating correctly, only 10% of the power meters will be defective. A vendor has just received a shipment of 14 power meters from the manufacturer and 5 have some form of defect. The vendor wants to test whether the proportion defective is actually greater than 10%. Use  = 0.05.

2. A small university library typically does a complete shelf inventory once every year. Because of new shelving rules, the head librarian believes it may be possible to save money by postponing the inventory, if it is not needed. In order to decide, the librarian selects, at random, 800 books from the library's collection and conducts a search for them. If the proportion of missing or mis-shelved books is less than 0.02 percent, the inventory will be postponed. Among the 800 books searched, 12 were either missing or mis-shelved. Conduct an appropriate hypothesis test and advise the librarian as to what to do. Use  = 0.05 and conduct the test by both the critical region method and by calculating a P-value.

3. Environmentalists strongly encourage people to carpool in order to conserve energy. Most cities have created an incentive for carpooling by designating certain highway traffic lanes as "car-pool only" (HOV or High Occupancy Vehicle lanes) during heavy traffic periods. To evaluate the effectiveness of this plan, engineers in one city monitored 2,000 randomly selected cars prior to establishing HOV lanes and 1,500 cars after the HOV lanes were established. Out of the sample of size 2,000, 652 cars had two or more passengers while out of the 1,500 cars, 576 cars had two or more passengers. Do the data indicate that the proportion of cars with multiple riders has increased as a result of the plan? Use  = 0.05 and conduct the test by both the critical region method and by calculating a P-value.

4. The normal distribution is used as an assumption in many statistical problems. Listed below is data that is to be used in a statistical study.

37.8 38.0 40.8 41.0 42.6 42.9
43.5 45.6 46.2 47.6 48.3 48.6
49.3 49.7 49.8 50.4 50.8 51.2
52.0 52.7 53.4 55.0 56.5 58.0
58.3 59.4 60.4 60.7 62.5 62.8

(a) Use the chi-square goodness-of-fit test to test that this data comes from a normal distribution. Use α = 0.05.
(b) Use the K-S test to test that this data comes from a normal distribution. Use α = 0.05.

5. The data below represents the time between arrivals (in seconds) at an ATM on a Friday afternoon.

0.04 0.06 0.09 0.11 0.26 0.27 0.38
0.45 0.46 0.48 0.59 0.67 0.72 0.78
0.79 0.88 0.96 1.11 1.12 1.19 1.39
1.56 1.60 1.78 1.87 1.89 1.96 1.98
1.99 2.09 2.34 2.41 2.80 2.89 2.93

(a) Use the chi-square goodness-of-fit test to test that this data comes from an exponential distribution. Use α = 0.10.
(b) Use the K-S test to test that this data comes from an exponential distribution. Use α = 0.10.

6. Consider a large population of families in which each family has three children. If the genders of the three children in any family are independent of one another, the number of male children in a randomly selected family will have a binomial distribution based on three trials. Suppose a random sample of 160 families yields the following results:

Number of
Male Children
0 1 2 3
Frequency 14 66 64 16

Test that the number of male children in a family composed of 3 children follows a binomial distribution with b (x; 3, 0.5) using α = 0.05.