Self Test 7 for Probability and Statistics

### Tests of Hypothesis and Significance

This test was constructed based on the Schaum's Outline Theory and Problems of Probability and Statistics by Murray R. Spiegel. If you need more review refer also to this outline.

Tests of Means and Proportions Using Normal Distributions

1. True False. The mean life time of a sample of 100 light bulbs produced by a certain company is computed to be 1550 hours with a standard deviation of 120 hours. If μ is the mean lifetime of all the bulbs produced by the company, test the hypothesis μ =1600 hours against the alternative hypothesis μ ≠1600 hours, using 5% level of significance.

a)Reject

(Contributed by Zauresh Atakhanova)

2. True False The mean service lives from a sample of 25 new, heavy-duty ore trucks from purchase to their first major repair is computed to be 3200 hours. The standard deviation is known at 250 hours. If m is the mean service time of all the trucks of that model, test the hypothesis m=3100 hours against the alternative hypothesis μ 3100 hours, using 5% level of significance.

We conclude that we must reject the null hypothesis at the 5% level of significance, i.e. that the mean time until major repair is not 3100 hours. (Contributed by James Golden)

3. True False. In problem 1 test the hypothesis m=1600 hours against the alternative m<1600, using a 5% level of significance.

a) Reject

(Contributed by James Golden)

4. True False Suppose that a random variable Y is such Y~ N(34.8,280). In addition suppose that you are unaware that E(Y) = b = 34.8, and that you wish to use a random sample of 60 observations to test.

Ho= b = 30 against

H1 = b > 30

If you use a %5 significance level, then the probability of making a Type I error is 0.987.

(Contributed by Carlos Roman)

## Tests Involving Differences of Means and Proportions

5. True False. A test was given to two classes consisting of 50 and 60 students respectively. In the first class the mean grade was 75 with standard deviation of 8. In the second class the mean grade was 78 with standard deviation of 7. Is the performance of the two classes significantly different at 1% and 5% level?

(a) No difference at both 1% and 5% significance level

(Contributed by Carlos Roman)

6. True False You have been monitoring crude oil production in Australia and Venezuela. Your have 600 days of data for Australia and 575 days for Venezuela. Suppose the mean production in barrels is 1200 thousand barrels per day (tbl/d) and 1300 (tbl/d) with a standard deviation of 180 and 150, respectively. Is the production of the two countries significantly different al 1% and 5% level? (Contributed by Maria Sanchez)

## Tests involving Student's t-distribution

7. True False. In the past a machine produced pieces of equipment with a mean thickness of 0.060 inches. To determine whether the machine is in the proper working condition, a sample of ten pieces of equipment is chosen for which the mean thickness is 0.063 and the standard deviation is 0.003 inches. You should reject the hypothesis that the machine is in the proper working condition using a level of significance of 0.05 but fail to reject at a 1% significance level.

(Contributed by Maria Sanchez)

8. True False A directional drilling tool is designed to build angles at a rate of 0.100 degrees/foot. To determine whether the machine is in the proper working condition, sixteen wells are sampled for which the sample mean build rate is 0.106 and the standard deviation is 0.012 degrees/foot. testing wheter your should expect more than a 0.100 build rate? using a level of significance 0.05 and 0.01.

After testing, the hypothesis that the machine is in the proper working condition you fail to reject H0 at the 5% and 1% significance levels.

(Contributed by James Golden)

9. True False In the past a machine produced pieces of iron with a mean thickness of 0.25 inches. To determine whether the machine is in proper working condition a sample of 15 pieces of iron are chosen for which the mean thickness is 0.28 and the standard deviation is 0.03 inches. Is the machine in the proper working condition using a level of significance of 0.05 and 0.01 we can reject al %% and fail to reject at 1% level of significance. (Contributed by Maria Sanchez)

10. True False In the past the standard deviation of weights of certain 40.0 gram packages filled by a machine was 0.25 grams. A random sample of 30 packages showed a standard deviation of 0.32 grams. The apparent increase in variability is significant at 0.05 level? (Contributed by Maria Sanchez)

## Tests involving the F distribution

11. True False A professor has two classes, X and Y. Class X had 16 students and class Y has 25 students. On the same test, although there was no significant difference in mean grades, class X had a standard deviation of 10 while class Y had a standard deviation of 13. We can conclude at 1% level of significance, that the variability of class Y is greater than that of X? (Contributed by Maria Sanchez)

## The chi-square test

12. True False. In 200 tosses of a coin, 116 heads and 84 tails were observed. Can we fail to reject testing the hypothesis that the coin is fair using a level of significance of 0.05. (Contributed by Maria Sanchez)

13. True False In the past a machine produced pieces of equipment with a mean thickness of 0.060 inches. To determine whether the machine is in the proper working condition, a sample of ten pieces of equipment is chosen for which the mean thickness is 0.063 and the standard deviation is 0.003 inches. A two-tailed test using a level of significance 0.05 is required to show that the machine is in the proper working condition. (Contributed by Herman Logsend)