Math 1070-003
Examples
1 March 2013
We will work together in class to answer the questions below. Use a different
sheet of paper to take notes, and show all of your work. We may not go over
all problems.
Name:
The original version of this handout contained two errors. They have been corrected in
bold-face type.
This question is related to Chapter 10 of your textbook.
1. You have two drawers of socks, labeled “A” and “B.” Each morning you choose one sock
from drawer A to wear on your right foot and one sock from drawer B to wear on your
left foot. Drawer A contains a green sock, a red sock, a blue sock and a black sock.
Drawer B contains two red socks, one white sock, and one black sock.
(a) Write out the sample space S for pairs of socks you can choose using this method,
and give the probability of each event.
(b) What is the probability that you will choose a matching pair of socks?
(c) What is the probability that your socks will not match?
(d) What is the probability that at least one of your socks will be red?
These questions are adapted from exercises in Chapter 14 of your textbook.
2. A laboratory scale is known to have a standard deviation of σ = 0.001 gram in repeated
weighings. Scale readings in repeated weighings are Normally distributed, with mean
equal to the true weight of the specimen. Three weighings of a specimen on this scale
give 3.412, 3.416, and 3.414 grams. What is a 90% confidence interval for the true weight
of this specimen?
3. What is a 99% confidence interval for the true weight of the specimen in question 2?
4. How many times would you need to weigh a particular specimen, using the scale from
question 2, in order to estimate its true weight with a margin of error less than .0001
grams with 99% confidence?
5. Successful hotel managers have personality characteristics thought of as feminine (such
as “compassionate”) as well as those thought of as masculine (such as “forceful”). The
Bem Sex-Role Inventory (BSRI) is a personality test that gives separate ratings for
female and male stereotypes, both on a scale of 1 to 7. The standard deviation of scores
on the BSRI for adult males is 0.78. A study attempted to answer the question “Do
hotel managers on average differ significantly in femininity score from men in general?”
A sample of 148 male general managers of three-star and four-star hotels had mean BSRI
femininity score 5.29. The mean score for the general male population is 5.19.
(a) State the null and alternative hypotheses in terms of the mean femininity score µ
for male hotel managers.
(b) Find the z test statistic (z-score).
(c) What is the P -value for your z?
(d) What do you conclude about male hotel managers?
6. A 95% confidence interval for a population mean is 31.5 ± 3.4. With a two-sided alternative, can you reject the null hypothesis that µ = 34 at the α = 0.05 significance level?
Why?
This question is adapted from the Mathematica exhibit at the Boston Museum of Science.
7. Due to rationing, a government mandate in a communist country requires that all loaves
of bread weigh 1 kg. Dough is weighed before it is shaped into loaves and baked,
so variation in the final weights can occur due to variations in rising and baking. It
is known that the weights of finished loaves are normally distributed, with standard
deviation 0.023 kilograms. A particular university’s dining hall receives a large weekly
delivery of bread. A professor from the Mathematics Department chooses an SRS of
100 loaves and weighs them using a very accurate scale. He calculates that the average
weight of the loaves is 0.892 kg. Is there evidence (at the α = 0.01 significance level)
that the bakery is producing loaves of bread that are smaller than the required 1 kg?
(a) State the null and alternative hypotheses.
(b) Find the z test statistic.
(c) What is the P -value for your z?
(d) What should the professor conclude?
These questions are adapted from the text Probability & Statistics for Engineering and Social
Sciences by J. L. Devore.
8. Lightbulbs of a certain type are advertised as having an average lifetime of 750 hours. It is
known that the lifetimes of these bulbs are Normally distributed with standard deviation
38.20 hours. A consumer reports group conducted a study to see if the average lifetime
was actually less than the advertised lifetime. If it is determined that the average lifetime
is less than 750 hours, the group will recommend that consumers not purchase the bulbs.
A random sample of 50 bulbs was selected, and the lifetime of each bulb was measured.
The average lifetime of the bulbs in the study was 738.44 hours. What conclusion can
you draw from the data at a significance level of 0.05?
Solution:
σ = 38.20, x = 738.44, n = 50
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We want to test the hypotheses:
H0 : µ = 750 vs. Ha : µ < 750
z=
738.44 − 750
38.20
√
50
=
738.44 − 750
= −2.1398
5.402
P = P (X ≤ −2.1398) = .0162
Since P < .05, the data is significant at the 0.05 level, and we conclude that the
average lifespan of these bulbs is most likely less than 750 hours.
9. What conclusion can you draw from the data in question 8 at a significance level of 0.01?
Which significance level and conclusion would you recommend?
Solution: Since P > 0.01, we would not conclude that the data is significant at the
0.01 level.
Since this is merely a consumer reports recommendation, not a medical situation or
a long-standing general belief, the first significance level is probably sufficient. The
P-value is not much larger than 0.01.
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