Download Math 116 – 05: Test #4 (Chapters 16 – 19) Name: Spring 2013

 120 Math 116 – 05: Test #4 (Chapters 16 – 19)
Name:
Spring 2013
Show work when possible to receive full credit for correct answers.
Give grammatically correct and clear explanations when asked to explain anything.
Problems 1 – 4 are worth 5 points each.
1. We have calculated a 95% confidence interval and would prefer for our next confidence
interval to have a smaller margin of error. In order to do this, we can:
I. increase the confidence level
II. decrease the confidence level
III. take a larger sample
IV. take a smaller sample
(a) I only
(b) II only
(c) III only
(d) IV only
(e) I and III
(f) I and IV
(g) II and III
(h) II and IV
2. A certain population is strongly skewed to the right. We want to estimate its mean, so we
will collect a sample. Which should be true if we use a large sample rather than a small one?
I. The distribution of our sample data will be closer to normal
II. The distribution of the sample means will be closer to normal
III. The variability of the sample means will be smaller
(a) I only
(b) II only
(c) III only
(d) I and III
(e) II and III
3. We have calculated a confidence interval based upon a sample of n = 160. Now we want to
get a better estimate with a margin of error only ¼ as large. We need a new sample with n at
least...
(a) 40
(b) 400
(c) 640
(d) 2560
(e) cannot be determined
4. Samples of size n are drawn from a normal population. For which value(s) of n would the
distribution of the sample means be normal?
(a) n = 5
(b) n = 15
(c) n = 25
(d) n = 35
(e) all of the above
5. A certain commuter must pass through five traffic lights on her way to campus and must stop
at each one that is red. She estimate the probability model for x = the number of red lights at
which she will be stopped as shown below.
(a) Find the probability that she must stop at exactly 2 red lights.
(b) Find the expected value (mean) and standard deviation of x.
(4 points)
(6 points)
x
0
1
2
3
4
5
P(x)
0.05
0.25
?
0.15
0.15
0.05
6. Your company bids for two contracts (each of which is worth $50000 to the company). The
probability you get the first contract is 0.8. If you get the first contract, the probability that
you also get the second is 0.2. If you do not get the first, the probability that you get the
second is 0.3. Let x = the amount of money your company gets from these two contracts.
Give the distribution (table of values, probabilities, mean and standard deviation) of x.
(10 points)
7. For each of the following scenarios, give the mean and standard deviation of the distribution
of the sample proportion (i.e. the p̂ distribution). Also state whether or not we can
legitimately assume the distribution is approximately normal. (7 points each)
(a) n = 40, p = 0.38
(b) n = 75, p = 0.12
8. It is generally believed that, when taking a certain medicine, 14% of people will develop an
itchy rash. What is the fewest number of people that could be sampled such that the p̂
distribution would be approximately normal in this scenario? (8 points)
9. For each of the following scenarios, give the mean and standard deviation of the distribution
of the sample mean (i.e. the x distribution). Also state whether or not we can legitimately
assume the distribution is approximately normal. (7 points each)
(a) m = 750, s = 84.5, n = 64
(b) m = 98.6, s = 3.8, n = 20
10. Boxes of Raspberry Crunch cereal contain a mean of 14.2 ounces with a standard deviation
of 0.5 ounce. The distribution of the contents of cereal boxes for this company is
approximately Normal. (6 points each)
(a) What is the probability that a single randomly selected box of this cereal contains
more than 14.82 ounces?
(b) What is the probability that in a randomly selected sample of 64 cereal boxes, the
average content is more than 14.82 ounces?
11. Information on a packet of seeds claims that the germination rate is 92%. What is the
probability that more than 95% of 160 randomly chosen such seeds will germinate? Be sure
to justify your work. (6 points)
12. A sociologist wants to estimate the percentage of Americans who are in favor of affirmative
action programs for minorities for admission to colleges and universities. Assuming there is
no previous data to consider, what sample size should be obtained if she wishes to construct a
95% confidence interval with an error of no more than 3 percentage points? (8 points)
13. In a random sample of 150 men, 82 of the men claimed to exercise regularly. Use the
information from this sample to construct a 97.8% confidence interval for the proportion of
all men that exercise regularly. Write a sentence interpreting the results. (9 points)
14. In a random sample of 180 women, about 64% of the women reported that they exercise
regularly. Use the information from this sample to construct a 98.5% confidence interval for
the proportion of all women that exercise regularly. Write a sentence interpreting the results.
(9 points)