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
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
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 =
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
- c) How likely is this difference if it is due to
- d) Should we accept random chance as the explanation
of the difference? What action should the company take?
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
- 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
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?
Std. Deviation: 18.2234
Std. Error Mean: 4.8704
Test Value = 300
Sig. (2-tailed): .785
Mean Difference: -1.3571
What belongs in df?
What is the value of t?
What do they conclude?