Download Math 116 – 05: Test #4 (Chapters 16 – 19) Name: Fall 2012 Show

 122 Math 116 – 05: Test #4 (Chapters 16 – 19)
Name:
Fall 2012
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 without losing any confidence. In order to do this,
we can:
I. change the z* - value to a smaller number
II. take a larger sample
III. take a smaller sample
(a) I only
(b) II only
(c) III only
(d) I and II
(e) I and III
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 sampling model of the sample means will be closer to normal
III. The variability of the sample means will be greater
(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 = 200. Now we want to
get a better estimate with a margin of error only one fifth as large. We need a new sample
with n at least...
(a) 40
(b) 240
(c) 450
(d) 1000
(e) 5000
4. Which is true about a 95% confidence interval based on a given sample?
I. The interval contains 95% of the population
II. Results from 95% of all samples will lie in the interval
III. The interval is narrower than a 98% confidence interval would be
(a) None
(b) I only
(c) II only
(d) III only
(e) II and III
5. A certain professor gives a weekly quiz with varying number of questions. The table below
gives the distribution of the variable x = # of questions on the quiz.
(a) Find the probability that the quiz has 30 questions.
(4 points)
(b) Find the expected value (mean) and standard deviation of x.
x
10
15
20
25
30
P(x)
0.05
0.20
0.50
0.15
?
(6 points)
6. A game is played with one 12-sided die. The game costs $10 to play. On the first roll, if you
roll a 5 you receive $75 and the game is over. If you do not roll a 5 on the first roll, then you
get a second roll. On the second roll, if you roll a 5 or 10, you get $50 and the game is over.
If you do not roll a 5 or 10 on the second roll, then you get nothing. Let x = the amount of
money your profit (not the amount won, but the amount you profit) when you play this game.
Give the probability model for the random variable x, and then find the expected value 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 = 60, p = 0.14
(b) n = 200, p = 0.38
8. It is generally believed that headaches as a side effect to a certain medicine affects about 18%
of people. 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 =1200, s =165, n = 64
(b) m = 98.6, s =1.2, n =16
10. Boxes of Raspberry Crunch cereal contain a mean of 13.4 ounces with a standard deviation
of 0.5 ounce. The distribution of the contents of cereal boxes is approximately Normal.
(6 points each)
(a) What is the probability that a single box of this cereal contains more than 13.62
ounces?
(b) What is the probability that in a case of 25 cereal boxes, the average content is more
than 13.62 ounces?
11. The President’s job approval rating is always a hot topic. Your local paper conducts a poll of
100 randomly selected adults to determine the President’s job approval rating. What is the
probability that in the sample, the President’s approval rating is below 50% given that his
actual approval rating is 54%? (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 4 percentage points? (8 points)
13. A random sample of 150 men found that 88 of the men 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. (10 points)
14. In a random sample of 200 women, 62% of the women reported that they exercise regularly.
Use the information from this sample to construct a 96.5% confidence interval for the
proportion of all women that exercise regularly. Write a sentence interpreting the results.
(10 points)