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### Problems: Testing Part 1

1. The mayor of a large city claims that the average age of persons arrested is 22.3 years. An investigative reporter doubts this claim, believing that the average age is younger. He takes a sample of 100 cases and finds that average age a bit less than 22.3. Is this conclusive proof that the mayor is a liar? Explain. (No need for any computation.)

2. Suppose that a library finds, by utilizing its computer records, that the mean number of pages in its books is 300 with a standard deviation of 60. If a student doing an assignment for a statistics class takes a random sample of 36 books, what is the probability of getting a sample mean that is more than 15 pages away from the true mean?

3. Suppose that a credit card company finds, using its computer records, that the average customer owes \$750 with a standard deviation of \$250. If a random sample of 225 customers is taken, what is the probability of getting a sample mean that is more than \$25 away from the true mean?

4. One area in which industry uses hypothesis testing is in quality control. For example, suppose that a manufacturer of candy bars wants its bars to weigh an average of 2 ounces. Because of wear, tear, and vibration, the machinery making the bars will occasionally make the bars too small or too big and will need adjustment. To see if the machinery is working satisfactorily, 25 bars are periodically checked and their average weight found.

a) Formulate the hypothesis and the alternative and state a decision rule (in terms of a t-value) if alpha = .05.
b) Suppose that the sample average is 2.036 ounces and the sample standard deviation is 0.090 ounces. How many standard errors is the sample mean away from the claimed mean?
c) How likely is this difference if it is due to random chance?
d) Should we accept random chance as the explanation of the difference? What action should the company take? Explain.

5. A manufacturer of lightbulbs wants to know whether its lightbulbs are better or worse than the lightbulbs of a competitor. The company's statistician knows that the company's lightbulbs last an average of 1550 hours.

a) What is the null hypothesis and what is the alternative?
H0:
HA:
b) Taking a random sample of 100 of the competitor's lightbulbs, he finds a mean lifetime of 1585 hours with a standard deviation of 150 hours. If alpha = .05, should he reject the null hypothesis? Why or why not?

6. A box contains four envelopes. One envelope has a \$50 bill, two envelopes have \$10 bills, and one has a blank piece of paper.

a) If you randomly pick an envelope from this box, what is the expected value of your pick?
b) Suppose you have to pay \$1.00 to pick an envelope from this box. You pay the \$1.00 and the envelope you pick has a blank piece of paper. Did you make a mistake in playing this game? Explain carefully.

7. Suppose a consumer group wants to test the claim that the average number of raisins in a brand of cereal is at least 300 per box. It takes a random sample of 100 boxes, counts the raisins, and finds that the standard deviation is 20 raisins. If they use alpha at .05, what range of sample means will convince them that the company's claim is inaccurate? Explain very carefully how you get your answer.

8. Suppose a consumer group doubts the claim that the average number of raisins in a brand of cereal is at least 300 per box. They take a sample of 14 and get the result below. What value belongs under t?

N: 14
Mean: 298.6429
Std. Deviation: 18.2234
Std. Error Mean: 4.8704
Test Value = 300
t: ____
df: ____
Sig. (2-tailed): .785
Mean Difference: -1.3571

What belongs in df?
What is the value of t?
What do they conclude?