Statistics
Review of Chapters 4, 5, and 6
1. Determine whether the random variable, x, is discrete or continuous.
(a) Let x represent the high temperature on a randomly selected day in July.
(b) Let x represent the number of people who attend a concert on a randomly selected performance.
2. Determine whether or not the distribution below is a probability distribution. If not, identify the
property that is not satisfied.
x
P(x)
0
0.15
1
0.35
2
0.25
3
0.15
4
0.10
5
0.05
3.
(a) Use the frequency distribution below to construct a probability distribution.
Number of Keys per Person in a Group of Students
Keys (x)
People (f)
1
2
3
4
5
6
4
15
10
5
4
2
Relative
Frequency
P(x)
(b) Sketch a probability histogram, and describe its shape.
(c) Find the mean and standard deviation of the probability distribution (using a calculator if you wish).
(d) Interpret the mean and standard deviation.
4. 37% of the adults in the US have type O+ blood. Blood samples are taken from a sample of 10 US
adults.
(a) For this binomial experiment, find the values of n, p, and q.
(b) Find the probability that 3 of the adults in the sample have type O+ blood. P(3) =
(c) Find the probability that three or fewer adults in the sample have type O+ blood. Also find the
probability that more than three adults in the sample have O+ blood. (Find
and
.)
(d) Give the mean, variance, and standard deviation of the distribution.
5. Use a calculator or the standard normal table to find the indicated area under the standard normal
curve. Also, shade the appropriate area under the curve.
(a) Find the area to the left of z = -1.24.
(b) Find the area to the right of z = 2.35.
(c) Find the area between z =
and z = 1.20.
6. Find the indicated probabilities (where z is the standard normal variable).
(a)
(b)
(c)
7. Random variable X is normally distributed with mean µ = 500 and standard deviation σ = 120. Find
the indicated probabilities.
(a)
(b)
(c)
8.
(a) Find
.
(b) Find the z-score such that the area to the left of z is 0.4575.
(c) Find the z-score such that the area to the right of z is 0.075.
(d) Find the z-scores such that the area between –z and z is 0.80.
9. Random variable X is normally distributed with mean µ = 500 and standard deviation σ = 120.
(a) Find the 25th percentile of this distribution.
(b) Find the value of x that cuts off the top 5%.
(c) Find the values of x that cut off the middle 98%.
10. The heights of males in a certain population are normally distributed with mean height 69 inches
and standard deviation 3.6 inches.
(a) Find the probability that a randomly selected male from this population is at least 72 inches tall.
(b) Find the probability that a randomly selected male from this population is shorter than 67 inches.
(c) Find the probability that a randomly selected male from this population is between 68 and 70 inches
tall.
(d) Find the 80th percentile of this distribution. Interpret this value.
(e) Find the shortest height that can be in the top 10% of heights in this population.
11. The heights of males in a certain population are normally distributed with mean height 69 inches
and standard deviation 3.6 inches. A random sample of 16 males is selected from this population.
(a) Give the mean and standard deviation of the sampling distribution of the sample mean.
(b) Find the probability that, in this sample of 16 males, the sample mean height is at least 70 inches.
(c) Find the probability that, in this sample of 16 males, the sample mean height is less than 67 inches.
Also, compare this probability with the result of problem 10b.
(d) Find the probability that, in this sample of 16 males, the sample mean is between 68 inches and 70
inches.
12. The mean weight of cereal in a box of a certain brand of corn flakes is claimed to be 15 ounces.
Suppose that the population standard deviation is 0.16 ounce. A random sample of 64 boxes is selected.
(a) Assuming that the population mean weight really is 15 ounces, describe the sampling distribution of
the sample mean (type of distribution, mean standard deviation).
(b) Suppose that, in this sample of 64 boxes, the sample mean weight is 14.92 ounces. Would this be
unusual if the claimed mean weight of 15 ounces is correct? Explain.
13. Suppose that, in reality, 48% of all voters in a state election favor a new tax. A random sample of
1600 potential voters is polled.
(a) Show that you can use a normal distribution to approximate the distribution of sample proportions.
(Check that
and
.)
(b) Find the mean and standard deviation of the distribution of sample proportions.
=
(c) Find the probability that more than 50% of the people in the sample favor the new tax.
)=
14. Use an appropriate procedure (ZInterval or TInterval) to construct a 95% confidence interval for the
population mean. Identify the procedure that you used.
(a)
(b)
15. Construct a 95% confidence interval for the population proportion.
(b)
16. In a random sample of 64 recent college graduates, the sample mean starting salary was found to
be $35,560 with a standard deviation of $1,250.
(a) Construct a 90% confidence interval for the population mean starting salary.
(b) Give the margin of error of the confidence interval.
(c) Interpret this confidence interval.
(d) If you were to construct a 99% confidence interval, would the margin of error be larger or smaller?
17. The following data represents the number of points scored by a basketball player in 10 randomly
selected games.
16
10
8
12
6
14
11
10
4
17
(a) Construct a 99% confidence interval for the population mean score.
(b) Give the margin of error of this confidence interval.
(c) Interpret this confidence interval.
18. In a survey of 700 parents of young children, 298 responded that they read to their children daily.
(a) Find the sample proportion of parents who read to their children daily.
(b) Construct a 95% confidence interval for the population proportion.
(c) Give the margin of error of this confidence interval.
(d) Interpret this confidence interval.
19. A large survey of older adults in the United States was taken in 2009. A summary of the survey
reported that a 95% confidence interval for the proportion of older adults who smoked was (0.081,
0.086).
(a) Find the sample proportion of smokers in the survey.
(b) Find the margin of error of the confidence interval.
(c) Interpret this confidence interval.
(d) In a similar survey done in 2007, a 95% confidence interval for the proportion of older adults who
smoked was (0.078, 0.102). Is it possible that the population proportion of older adults who smoked in
2007 was actually smaller than corresponding proportion in 2009? Explain.
20. You are trying to estimate the mean distance that college students commute to school. Find the
minimum sample size that would be needed to construct a 95% confidence interval for the mean with a
margin or error of 2 miles. (σ = 4 miles.)
21. You wish to estimate the population proportion of college students who work at least part time.
Find the minimum sample size that you would need to construct a 95% confidence interval for the
population proportion with a margin of error of 3% if
(a) no prior estimates are available.
(b) in an earlier study, the sample proportion was 67%.