MATH 170 Review problems for the Final Exam (Chapters 8, 9 &10)
Ch. 8 Sampling Distributions
Distribution of the Sample Mean
(a) Distribution of the Sample Mean for Samples Obtained from Normal Populations
The average score of all golfers for a particular course has a normal distribution with mean of 68
and a standard deviation of 5. Suppose 10 golfers played the course today. Find the probability
that the average score of the 10 golfers exceeded 69.
Answer: 0.2635
(b) Distribution of the Sample Mean for Samples Obtained from a Population That Is Not
Normal
1. The number of violent crimes committed in a day possesses a distribution with a mean of 2.5
crimes per day and a standard deviation of 3 crimes per day. A random sample of 100 days was
observed, and the mean number of crimes for the sample was calculated. Describe the sampling
distribution of the sample mean.
Answer: Approximately normal with mean = 2.5 and standard deviation = 0.3,due to the large
sample size.
2. The owner of a convenience store has determined that their daily revenue has mean $7200 and
standard deviation $1200. The daily revenue totals for the next 30 days will be monitored. What is
the probability that the mean daily revenue for the next 30 days will be between $7000 and $7500?
Answer: 0.7333
Distribution of the Sample Proportion
Smith is a weld inspector at a shipyard. He knows from keeping track of good and substandard
welds that for the afternoon shift 5% of all welds done will be substandard. If Smith checks 300 of
the 7500 welds completed that shift, what is the probability that he will find between 10 and 20
substandard welds?
Answer: 0.8132
Ch. 9 Estimating the Value of a Parameter Using Confidence Intervals
Compute a Point Estimate of the Population Mean
The grade point averages for 10 randomly selected students in a statistics class with 125 students
are listed below. What can you say about the mean score μ of all 125 students?
3.1 2.3 2.7 1.8 3.2 3.0 2.0 3.4 2.2 2.6
Answer: 2.63
Construct and Interpret a Confidence Interval about the Population Mean – Z
Interval
1. Compute the critical value zα/2 that corresponds to a 95% level of confidence.
Answer: 1.96
2. In a sample of 10 randomly selected women, it was found that their mean height was 63.4
inches. From previous studies, it is assumed that the standard deviation, σ, is 2.4. Construct the
95% confidence interval for the population mean.
Answer: (61.9, 64.9)
3. Suppose a 95% confidence interval for μ turns out to be (120, 300). To make more useful
inferences from the data, it is desired to reduce the width of the confidence interval. Which of the
following will result in a reduced interval width?
A) Increase the sample size. B) Decrease the confidence level. C) Increase the sample size and
decrease the confidence level. D) All of the choices will result in a reduced interval width.
Answer: A
4. True or False: A confidence interval with a 95% level of confidence means that the parameter
will be in the interval for 95 out of 100 samples of the same size.
A) True
5. Suppose a 99% confidence interval for μ turns out to be (190, 250). Based on the interval, do
you believe the average is equal to 290?
A) No, and I am 99% sure of it.
Confidence Intervals about a Population Mean in Practice Where the Pop Std Dev
Is Unknown – T Interval
1. How much money does the average professional football fan spend on food at a single football
game? That question was posed to 10 randomly selected football fans. The sampled results show
that sample mean and standard deviation were $18.00 and $3.30, respectively. Use this
information to create a 95% confidence interval for the mean.
Answer: 18 ± 2.262(3.30/
10 )
2. The grade point averages for 10 randomly selected high school students are listed below.
Assume the grade point averages are normally distributed.
2.0 3.2 1.8 2.9 0.9 4.0 3.3 2.9 3.6 0.8
Find a 98% confidence interval for the true mean.
Answer: (1.55, 3.53)
Confidence Intervals about a Population Proportion – Population Z Interval
Many people think that a national lobby’s successful fight against gun control legislation is
reflecting the will of a minority of Americans. A random sample of 4000 citizens yielded 2250
who are in favor of gun control legislation. Estimate the true proportion of all Americans who are
in favor of gun control legislation using a 90% confidence interval.
Answer: .5625 ± .0129
Ch. 10 Testing Claims Regarding a Parameter
The Language of Hypothesis Testing
1. The mean age of bus drivers in Chicago is 50.1 years. Identify the type I and type II errors for
the hypothesis test of this claim.
Answer: type I: rejecting H0: μ = 50.1 when μ = 50.1
type II: failing to reject H0: μ = 50.1 when μ ≠ 50.1
2. Suppose you are using α = 0.05 to test the claim that μ > 4 using a P-value. You are given the
sample statistics n = 50, x = 4.3, and σ = 1.2. Find the P-value.
Answer: 0.0384
3. If the level of significance is 0.05, and the p-value is 0.043, the decision would be to
A) Reject H0
B) Fail to reject H0
Answer: A
Testing Claims about a Population Mean Assuming the Population Standard
Deviation Is Known (Or Using the sample Standard Deviation when the sample
size is large) – Z Test
A fast food outlet claims that the mean waiting time in line is less than 3 minutes. A random
sample of 60 customers has a mean of 2.9 minutes. Suppose the standard deviation is known to be
0.6 minute. If α = 0.05, test the fast food outlet’s claim using P-values.
Answer: P-value = 0.0985, P > α, fail to reject H0; There is not sufficient evidence to support the
claim that the mean waiting time is less than 3 minutes.
Testing Claims about a Population Mean in Practice – T Test
A local group claims that the police issue at least 60 speeding tickets a day in their area. To prove
their point, they randomly select two weeks. Their research yields the number of tickets issued for
each day. The data are listed below. At α = 0.01, test the group’s claim.
70 48 41 68 69 55 70 57 60 83 32 60 72 58
Answer: x = 60.21, s = 13.43; critical value t0 = 2.650; standardized test statistic ≈ 0.060; fail to
reject H0; There is not sufficient evidence to reject the claim.
Testing Claims about a Population Proportion – Population Z Test
Fifty-five percent of registered voters in a congressional district are registered Democrats. The
Republican candidate takes a poll to assess his chances in a two-candidate race. He polls 1200
potential voters and finds that 621 plan to vote for the Democratic candidate. Does the Republican
candidate have a chance to win? Use α = 0.05.
Answer: critical value z0 = 1.645; standardized test statistic ≈ 1.21; fail to reject H0; There is not
sufficient evidence to support the claim p > 0.5. The Republican candidate has no chance.