Problems: Testing Part 2
9. Below is sample data from one of the popular computer statistics programs:
One-Sample Statistics
N: 100
Mean: 49.0509
Std. Deviation: 5.15965
Std. Error Mean: .51596
- a) How big is the sample size? ________How big is the
population?
b) Suppose that someone made a claim that the true mean of the population from which this sample was drawn is 50. You want to test this claim to see if it looks reasonable. How will you compute your test statistic?
c) Does the claim look like it may be true, or does it look implausible? Explain.
d) What is a 95% confidence interval for the true mean?
10. A company has a hair tonic that it claims will increase hair growth. A group of skeptics decides to test this claim. They know that normally hair grows .2 mm a day with a standard deviation of .03 mm. (The numbers are made up for this problem.)
- a) What is the hypothesis and what is the alternative?
- b) They plan to take a sample of 16 people and to test this product on them. In stating the decision rule, should they use the normal or t tables? (Be careful.) Explain.
- c) State a decision rule (set alpha = .05.)
- d) Their sample mean is .22 mm. What should they conclude?
- e) Have they either proven or disproven the claim? What do your results mean?
11. An established variety of squash yields 2200 pounds per square acre. A newly-developed variety is planted on nine acres and yields 2500 pounds per acre with a standard deviation of 120 pounds. Test the hypothesis that the old variety yields as much as the new one against the hypothesis that the new variety yields more. Let alpha = .05.
12. A company supplying tiles to a contractor claims that its tiles weigh an average of 50 ounces, no more or no less. A random sample of 64 tiles weighs 50.32 ounces with a standard deviation of 1.20 ounces.
- a) If we establish alpha= .05, would you accept or
reject the company's claim? Explain.
b) Would your answer change if alpha = .01? Explain.
13. The Acme Pillow Company stuffs pillow with horsefeathers. The average weight of one of its pillows is 22 oz. with a standard deviation of 1.5 oz.
- a) What is the probability that one pillow selected at random will weigh 21 oz or less assuming that weight is distributed normally?
- b) A random sample of 25 pillows will be taken. What is the probability that the average weight of these pillows will be 21 oz. or less? Why, intuitively, is your answer different than the answer in part a?
- c) The company wants to readjust pillow-stuffing machines whenever pillows are on the average too small or too big (different from 22 oz.). Unnecessary adjustment is costly because it interrupts production. Explain carefully what the company should do if a random sample of 25 pillows has an average weight of 21 oz. (Make whatever assumptions you need, but explain them.)
14. In a study of economic education and attitudes toward free enterprise, two researchers examined the change in attitude of a large group of students with a questionnaire. The questionnaire was given both before and after a series of seminars on free-enterprise economics. The difference in means on the two tests was 1.02 with a standard error of .803, which gives a computed t-value of 1.27. Should the researchers decide that the seminars were effective in changing student attitudes toward free enterprise if alpha = .05? Explain the reasoning of your answer. (Data from an article in the Journal of Economic Education, Spring, 1978)
15. A manufacturer claims that its boxes of tissue paper contain 200 tissues. A consumer protection agency decides to check this claim with the thought of bringing a lawsuit if the evidence is strong enough that the company is lying.
- a) What is the hypothesis and what is the alternative? Is this a one-tailed or two-tailed test?
- b) Because bringing a lawsuit is costly, it decides to set alpha equal to .001. What does this mean?
- c) It takes a random sample of 100 boxes and finds that the sample standard deviation is 8 tissues. What is the critical or rejection region?
- d) Suppose alpha is set at .05. Would it start court action if x-bar =198.5?
16. A producer of birdseed wants its bags of birdseed to contain and average of 100.2 pounds. It finds that when its filling machine is working properly, some bags contain a bit more than 100.2 pounds and some contain a bit less. In fact, the standard deviation is .4 pounds.
If the weight of the bags follow a normal distribution and the machine is working properly, what percentage of them will contain more than 100 pounds?
The company takes random samples of bags and weighs them to see if the machine is working properly. The last group of 25 bags weighed an average of 99.9 pounds. Should the company conclude that the machine is filling the bags properly and the .3-pound difference is just random chance, or should it conclude that the machinery needs to be reset? Explain with words and calculations.
17. Economists are interested in how people value time. In an experiment, people were given gift certificates worth $7.00 for a record store. Some of them were given gift certificates that became valid in one week, and they were given an opportunity to trade it for a bigger certificate that became valid in four weeks. The average increase in value needed to get people to make this trade was $1.09. Others were given certificates that became valid in four weeks, and given an opportunity to trade them for certificates that became valid in only one week. The average decrease in value that they would accept to make this trade was $.25. They thus got two measures of what the delay of three weeks was worth. The question they had is whether this difference was real or whether was it random. Here is the table that summarizes the results. (Source: Richard Thaler, The Winner's Curse, p 101.)
Table 8-1 Mean Amounts to Speed-Up and Delay Consumption ($7 Record Store Gift Certificate)
Time Interval
Delay
Speed-Up
Significance
1 week versus 4 weeks
$1.09
$.25
.001
4 weeks versus 8 weeks
$.84
$.37
.005
1 week versus 8 weeks
$1.76
$.52
.001
Source: Lowensteing (1988)
Standard economics suggests that the value of an earlier coupon compared to a later one should not depend on which way the trade goes, from earlier to later or from later to earlier. Do these results support that assumption? Explain carefully.
18. Suppose that the diameters of oranges from a grove of orange trees is normally distributed. We wish to test the hypothesis that the mean diameter is three inches against the hypothesis that it is not equal to three inches. We are fairly certain that the true mean is three inches, so we do not want to abandon that assumption unless there is strong evidence against it, so we set the probability of Type I error at 5%. From the sample of 100 we find a standard deviation of one inch. When will reject the hypothesis that the true mean is three inches?
b) If the true mean is actually 3.2 inches, what is the probability that we will make Type II error by concluding that the true mean is three inches?
19. The output below comes from a statistical program:
One Sample Statistics: N Mean Std. Deviation Std Error Mean Variable 1
37
31.027
15.5768
2.5608
Test Value =30 t df sig (2-tailed) Mean Difference 95% Confidence Interval of the Difference Lower Upper Variable 1
.401
36
.691
1.0270
-4.1666
6.2206
- a) What is the hypothesis being tested?
- b) How big was the sample?
- c) What was the sample mean?
- d) How was the t-value computed and what does it mean?
- e) What conclusion would one draw from the significance of this test? Explain in several sentences.
20. The average SAT verbal score nationally in 1978 was 429 with a standard deviation of 110. Suppose we want to know whether students at our local college are different from this average.
- a) What is the hypothesis or claim that we initially presume to be true?
- b) What is the alternative hypothesis or claim?
- c) If a sample of 25 students yields an average of 394, what should we conclude if we set alpha = .05?
- d) Suppose we did not know the national standard deviation but used the standard deviation from the sample, which is 75. How does this change the procedure?
21. A manufacturer of batteries wants to know whether its batteries are better or worse than the batteries of a competitor. The company's statistician knows that the company's batteries last an average of 1640 hours.
- a) What is the null hypothesis (claim) and what is the alternative?
- H0:
HA: - b) Taking a random sample of 100 of the competitor's batteries, he finds a mean lifetime of 1605 hours with a standard deviation of 150 hours. If alpha = .05, should he reject the null hypothesis (claim)? Why or why not?
22. We have discussed how statistical inference resembles a trial by jury. Let's work though a couple problems to illustrate this, using some simple examples.
a) Suppose you have a die and you want to test it to see if it is a fair die, with equal chances of coming up with an even number (2, 4, 6) or an odd number (1, 3, 5). You toss the die 20 times. Using the table below, what is the probability that you will get exactly ten even tosses?
How many or how few even tosses would you have to get before you would conclude that the die was not fair? Explain. Put this in terms of a starting hypothesis and reasonable doubt.
20 trials, probability of success on any one trial = .5
Successes:
0
1
2
3
4
5
6
7
8
9
10Probability of n successes:
9.536743164e-7
1.907348633e-5
1.811981201e-4
1.087188721e-3
4.620552063e-3
0.0147857666
0.0369644165
0.073928833
0.1201343536
0.1601791382
0.17619705211
12
13
14
15
16
17
18
19
200.1601791382
0.1201343536
0.073928833
0.0369644165
0.0147857666
4.620552063e-3
1.087188721e-3
1.811981201e-4
1.907348633e-5
9.536743164e-7
b) A multiple choice test question has 20 questions on it, and each question has four possible answers.
If a person guessed at random, what is the probability that he or she would get exactly 6 right? (Use the table below.)
You are quite sure that John knows absolutely nothing and just guesses randomly on the test. How many questions would John have to get right before you would conclude he was not a complete knucklehead? Explain how you reach your decision.
20 trials, probability of success on any one trial = .25
Successes:
0
1
2
3
4
5
6
7
8
9
10Probability of n successes:
3.171211939e-3
0.0211414129
0.0669478076
0.1338956152
0.1896854549
0.2023311519
0.1686092932
0.1124061955
0.0608866892
0.0270607508
0.009922275311
12
13
14
15
16
17
18
19
203.006750085e-3
7.516875212e-4
1.54192312e-4
2.569871867e-5
3.426495823e-6
3.569266482e-7
2.799424692e-8
1.55523594e-9
5.456968211e-11
9.094947018e-13
23. The output below comes from a statistical program:
One Sample Statistics: N Mean Std. Deviation Std Error Mean Age
115
66.2563
15.2338
1.4206
Test Value =70 t df sig (2-tailed) Mean Difference 95% Confidence Interval of the Difference Lower Upper Age
-2.635
144
.010
-3.7437
-6.5578
-0.9295
- a) What is the hypothesis being tested?
- b) How big was the sample?
- c) What was the sample mean?
- d) How was the Std Error Mean computed?
- d) How was the t-value computed and what does it mean?
- e) What conclusion would one draw from the significance of this test? Explain in several sentences.
- f) Below are other results from another test with
another set of data. Fill in the missing boxes.
One Sample Statistics: N Mean Std. Deviation Std Error Mean Var2
73
230.1861
5.1716
_____
Test Value =229 t df sig (2-tailed) Mean Difference 95% Confidence Interval of the Difference Lower Upper Var2
_____
_____
.054
1.1861
-.0205
_____
24. "Patients treated with LUNESTA demonstrated statistically significant improvement (p<0.01) compared with placebo in patient-reported measures of sleep latency (onset of sleep), wake time after sleep onset (WASO; a sleep maintenance measurement of the amount of time spent awake after initially falling asleep) and total sleep time for each week of the study. "
Explain what this quotation is saying to someone who has not had a course in statistics. In particular, what is the importance of the term "statistically significant" and what does the p<0.01 mean?
- 25. A company buys a large shipment of ducksoup from a supplier. The mean weight of the packages is supposed to be 400 grams, but a sample of 100 packages has a mean weight of 396 grams with a standard deviation of 10 grams. The supplier says that the four-gram difference is probably due to random chance and that a different sample would probably be at least 400 grams. Do you agree that random chance is a reasonable explanation for the discrepancy? Explain carefully.
- 26. The Ed Gien Bakery specializes in ladyfingers. The size of a ladyfinger is normally distributed with an average length of six inches and a standard deviation of .2 inches.
- a) What is the probability that one ladyfinger taken at random will be longer then 6.1 inches?
- b) What is the probability that the average length of a random sample of 16 will be greater than 6.1? Why intuitively is your answer different in part b than in part a?
- c) Suppose Mr. Gien wants to know if he is cutting his ladyfingers too long. If he wants the average length to be six inches and he finds that a sample of 16 averages 6.1 inches, should he take action to correct his procedures? Explain carefully, telling how the levels of type 1 and type 2 error will influence his decision.
27. When a firm hires, it can make two mistakes. It can
hire someone who turns out to be a "bad" employee, or it can
pass over someone who would have been a "good" employee.
These mistakes are similar to the mistakes a jury can make
in a criminal trial and to Type I and Type II errors in
statistical hypothesis testing. We set up statistical
hypothesis testing so that we attack the claim. There are
two claims that we can use in hiring: the candidate is good
and we should hire unless there is reason to not hire, or
the candidate is bad and we should not hire unless there is
good reason to hire. Which should be the default position
if:
a) supply is large and training is expensive?
b) supply is small and training is inexpensive?
c) supply is small and training is expensive?
28. An official with the Federal Energy Administration believes that motorists are averaging more than 70 miles per hour on the interstate highways. As a result, he maintains that a more strict enforcement of speed limits would save a considerable amount of gasoline. To test his hunch, he plans to measure the speed of a large sample of motorists.
- a) How would he set up his hypothesis and the alternative?
- b) If he decides to set alpha, the risk of type 1 error, at .05, what will his decision rule be?
- c) If he measures the speed of 144 motorists and finds the sample mean = 69.6, what will he decide?
29. Below is an except from a book on pyramid power, the idea that pyramids have special powers. (For a larger image, click it.)
- a) What hypothesis is being tested in the reading
above, and what is the alternative?
b) On the basis of the evidence, is the conclusion in the last paragraph justified? Explain.