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STT 315 Practice Problems III for Sections 5.1 - 7.6
For your convenience open-end quections 70 - 85 were converted to bimodal so you can see
solutions.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Answer the question True or False.
1) In most situations, the true mean and standard deviation are unknown quantities that have to be
estimated.
A) True
B) False
1)
2) The probability of success, p, in a binomial experiment is a parameter, while the mean and
standard deviation, µ and , are statistics.
A) True
B) False
2)
3) The sample mean, x, is a statistic.
A) True
3)
B) False
4) The term statistic refers to a population quantity, and the term parameter refers to a sample quantity.
A) True
B) False
4)
5) Sample statistics are random variables, because different samples can lead to different values of the
sample statistics.
A) True
B) False
5)
6) If x is a good estimator for µ, then we expect the values of x to cluster around µ.
A) True
B) False
6)
7) The sampling distribution of a sample statistic calculated from a sample of n measurements is the
probability distribution of the statistic.
A) True
B) False
7)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
8) Consider the population described by the probability distribution below.
x
p(x)
2
.2
5
.5
7
.3
The random variable x is observed twice. The observations are independent. The different
samples of size 2 and their probabilities are shown below.
Sample
2, 2
2, 5
2, 7
Probability
.04
.10
.06
Sample
5, 2
5, 5
5, 7
Probability
.10
.25
.15
Sample
7, 2
7, 5
7, 7
Find the sampling distribution of the sample mean x.
1
Probability
.06
.15
.09
8)
9) Consider the population described by the probability distribution below.
x
p(x)
a.
3
.1
5
.7
9)
7
.2
Find µ.
b. Find the sampling distribution of the sample mean x for a random sample of n = 2
measurements
from the distribution.
c.
Show that x is an unbiased estimator of µ.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Answer the question True or False.
10) The ideal estimator has the greatest variance among all unbiased estimators.
A) True
B) False
10)
11) The minimum-variance unbiased estimator (MVUE) has the least variance among all unbiased
estimators.
A) True
B) False
11)
12) If no estimate of p exists when determining the sample size for a confidence interval for a
proportion, we can use .5 in the formula to get a value for n.
A) True
B) False
12)
Solve the problem.
13) The Central Limit Theorem states that the sampling distribution of the sample mean is
approximately normal under certain conditions. Which of the following is a necessary condition for
the Central Limit Theorem to be used?
A) The population size must be large (e.g., at least 30).
B) The population from which we are sampling must not be normally distributed.
C) The population from which we are sampling must be normally distributed.
D) The sample size must be large (e.g., at least 30).
14) The number of cars running a red light in a day, at a given intersection, possesses a distribution
with a mean of 1.7 cars and a standard deviation of 5. The number of cars running the red light was
observed on 100 randomly chosen days and the mean number of cars calculated. Describe the
sampling distribution of the sample mean.
A) shape unknown with mean = 1.7 and standard deviation = 5
B) shape unknown with mean = 1.7 and standard deviation = 0.5
C) approximately normal with mean = 1.7 and standard deviation = 0.5
D) approximately normal with mean = 1.7 and standard deviation = 5
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
15) Suppose a random sample of n = 64 measurements is selected from a population with
mean µ = 65 and standard deviation = 12. Find the values of µx and x .
2
15)
13)
14)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
16) The weight of corn chips dispensed into a 14-ounce bag by the dispensing machine has been
identified as possessing a normal distribution with a mean of 14.5 ounces and a standard deviation
of 0.2 ounce. Suppose 100 bags of chips are randomly selected. Find the probability that the mean
weight of these 100 bags exceeds 14.6 ounces.
A) .1915
B) .3085
C) .6915
D) approximately 0
16)
17) The average score of all golfers for a particular course has a mean of 66 and a standard deviation of
3.5. Suppose 49 golfers played the course today. Find the probability that the average score of the
49 golfers exceeded 67.
A) .0228
B) .3707
C) .4772
D) .1293
17)
18) A random sample of n = 400 measurements is drawn from a binomial population with probability
of success .21. Give the mean and the standard deviation of the sampling distribution of the sample
18)
^
proportion, p.
A) .79; .02
B) .21; .008
C) .21; .02
D) .79; .008
19) Suppose a random sample of n measurements is selected from a binomial population with
probability of success p = .32. Given n = 400, describe the shape, and find the mean and the
19)
^
standard deviation of the sampling distribution of the sample proportion, p.
A) approximately normal; 0.32, 0.0005
B) skewed right; 128, 9.33
C) skewed right; 0.32, 0.023
D) approximately normal; 0.32, 0.023
20) A study was conducted to determine what proportion of all college students considered themselves
as full-time students. A random sample of 300 college students was selected and 210 of the
students responded that they considered themselves full-time students. Which of the following
would represent the target parameter of interest?
A) p
B) µ
20)
21) What is z /2 when
A) 1.96
21)
= 0.02?
B) 2.575
C) 1.645
D) 2.33
22) What is the confidence level of the following confidence interval for µ?
x ± 2.33
A) 98%
22)
n
B) 233%
C) 78%
D) 67%
23) A 90% confidence interval for the mean percentage of airline reservations being canceled on the
day of the flight is (1.1%, 3.2%). What is the point estimator of the mean percentage of reservations
that are canceled on the day of the flight?
A) 2.15%
B) 1.05%
C) 2.1%
D) 1.60%
23)
24) A random sample of 250 students at a university finds that these students take a mean of 15.3 credit
hours per quarter with a standard deviation of 1.6 credit hours. Estimate the mean credit hours
taken by a student each quarter using a 95% confidence interval. Round to the nearest thousandth.
A) 15.3 ± .010
B) 15.3 ± .157
C) 15.3 ± .013
D) 15.3 ± .198
24)
3
25) The director of a hospital wishes to estimate the mean number of people who are admitted to the
emergency room during a 24-hour period. The director randomly selects 49 different 24-hour
periods and determines the number of admissions for each. For this sample, x = 17.2 and s2 = 25.
25)
26) A 90% confidence interval for the average salary of all CEOs in the electronics industry was
constructed using the results of a random survey of 45 CEOs. The interval was ($110,389, $128,192).
Give a practical interpretation of the interval.
A) We are 90% confident that the mean salary of the sampled CEOs falls in the interval $110,389
to $128,192.
B) 90% of all CEOs in the electronics industry have salaries that fall between $110,389 to
$128,192.
C) We are 90% confident that the mean salary of all CEOs in the electronics industry falls in the
interval $110,389 to $128,192.
D) 90% of the sampled CEOs have salaries that fell in the interval $110,389 to $128,192.
26)
27)
27)
Estimate the mean number of admissions per 24-hour period with a 90% confidence interval.
A) 17.2 ± .643
B) 17.2 ± .168
C) 17.2 ± 1.175
D) 17.2 ± 5.875
Find the value of t0 such that the following statement is true: P(-t0
A) 2.2821
B) 1.833
C) 3.250
t t0 ) = .99 where df = 9.
D) 2.262
28) Let t0 be a specific value of t. Find t0 such that the following statement is true:
P(t t0 ) = .05 where df = 20.
A) -1.725
B) 1.725
C) -1.729
29) Let t0 be a specific value of t. Find t0 such that the following statement is true:
P(t t0 ) = .01 where df = 20.
A) -2.528
B) 2.528
C) -2.539
28)
D) 1.729
29)
D) 2.539
30) Private colleges and universities rely on money contributed by individuals and corporations for
their operating expenses. Much of this money is invested in a fund called an endowment, and the
college spends only the interest earned by the fund. A recent survey of eight private colleges in the
United States revealed the following endowments (in millions of dollars): 88.2, 52.1, 233.3, 480.6,
116.6, 179.1, 110.2, and 221.7. What value will be used as the point estimate for the mean
endowment of all private colleges in the United States?
A) 8
B) 211.686
C) 185.225
D) 1481.8
30)
31) You are interested in purchasing a new car. One of the many points you wish to consider is the
resale value of the car after 5 years. Since you are particularly interested in a certain foreign sedan,
you decide to estimate the resale value of this car with a 95% confidence interval. You manage to
obtain data on 17 recently resold 5-year-old foreign sedans of the same model. These 17 cars were
resold at an average price of $12,420 with a standard deviation of $700. What is the 95% confidence
interval for the true mean resale value of a 5- year-old car of this model?
A) 12,420 ± 2.120(700/ 16)
B) 12,420 ± 2.110(700/ 17)
C) 12,420 ± 1.960(700/ 17)
D) 12,420 ± 2.120(700/ 17)
31)
4
32) You are interested in purchasing a new car. One of the many points you wish to consider is the
resale value of the car after 5 years. Since you are particularly interested in a certain foreign sedan,
you decide to estimate the resale value of this car with a 95% confidence interval. You manage to
obtain data on 17 recently resold 5-year-old foreign sedans of the same model. These 17 cars were
resold at an average price of $12,830 with a standard deviation of $800. Suppose that the interval is
calculated to be ($12,418.66, $13,241.34). How could we alter the sample size and the confidence
coefficient in order to guarantee a decrease in the width of the interval?
A) Keep the sample size the same but increase the confidence coefficient.
B) Increase the sample size but decrease the confidence coefficient.
C) Decrease the sample size but increase the confidence coefficient.
D) Increase the sample size and increase the confidence coefficient.
32)
33) A marketing research company is estimating the average total compensation of CEOs in the service
industry. Data were randomly collected from 18 CEOs and the 98% confidence interval for the
mean was calculated to be ($2,181,260, $5,836,180). Explain what the phrase "98% confident" means.
A) 98% of the sample means from similar samples fall within the interval.
B) 98% of the similarly constructed intervals would contain the value of the sample mean.
C) 98% of the population values will fall within the interval.
D) In repeated sampling, 98% of the intervals constructed would contain µ.
33)
34) A marketing research company is estimating the average total compensation of CEOs in the service
industry. Data were randomly collected from 18 CEOs and the 97% confidence interval for the
mean was calculated to be ($2,181,260, $5,836,180). What additional assumption is necessary for
this confidence interval to be valid?
A) None. The Central Limit Theorem applies.
B) The sample standard deviation is less than the degrees of freedom.
C) The distribution of the sample means is approximately normal.
D) The population of total compensations of CEOs in the service industry is approximately
normally distributed.
34)
35) A marketing research company is estimating the average total compensation of CEOs in the service
industry. Data were randomly collected from 18 CEOs and the 99% confidence interval for the
mean was calculated to be ($2,181,260, $5,836,180). What would happen to the confidence interval
if the confidence level were changed to 98%?
A) The interval would get wider.
B) There would be no change in the width of the interval.
C) The interval would get narrower.
D) It is impossible to tell until the 98% interval is constructed.
35)
36) To help consumers assess the risks they are taking, the Food and Drug Administration (FDA)
publishes the amount of nicotine found in all commercial brands of cigarettes. A new cigarette has
recently been marketed. The FDA tests on this cigarette yielded mean nicotine content of 26.7
milligrams and standard deviation of 2.4 milligrams for a sample of n = 9 cigarettes. Construct a
98% confidence interval for the mean nicotine content of this brand of cigarette.
A) 26.7 ± 2.394
B) 26.7 ± 2.317
C) 26.7 ± 2.257
D) 26.7 ± 2.457
36)
5
37) A marketing research company is estimating which of two soft drinks college students prefer. A
random sample of 151 college students produced the following confidence interval for the
proportion of college students who prefer drink A: (.344, .494). Is this a large enough sample for this
analysis to work?
A) Yes, since n = 151 (which is 30 or more).
^
37)
^
B) Yes, since both np 15 and nq 15.
C) No.
D) It is impossible to say with the given information.
38) A marketing research company is estimating which of two soft drinks college students prefer. A
random sample of n college students produced the following 90% confidence interval for the
proportion of college students who prefer drink A: (.406, .586). Identify the point estimate for
estimating the true proportion of college students who prefer that drink.
A) .496
B) .406
C) .586
D) .09
38)
39) A Florida newspaper reported on the topics that teenagers most want to discuss with their parents.
The findings, the results of a poll, showed that 46% would like more discussion about the family's
financial situation, 37% would like to talk about school, and 30% would like to talk about religion.
These and other percentages were based on a national sampling of 549 teenagers. Using 99%
reliability, can we say that more than 30% of all teenagers want to discuss school with their
parents?
A) Yes, since the value .30 falls inside the 99% confidence interval.
B) No, since the value .30 is not contained in the 99% confidence interval.
C) No, since the value .30 is not contained in the 99% confidence interval.
D) Yes, since the values inside the 99% confidence interval are greater than .30.
39)
40) A university dean is interested in determining the proportion of students who receive some sort of
financial aid. Rather than examine the records for all students, the dean randomly selects 200
students and finds that 118 of them are receiving financial aid. Use a 95% confidence interval to
estimate the true proportion of students who receive financial aid.
A) .59 ± .068
B) .59 ± .005
C) .59 ± .002
D) .59 ± .474
40)
41) A university dean is interested in determining the proportion of students who receive some sort of
financial aid. Rather than examine the records for all students, the dean randomly selects 200
students and finds that 118 of them are receiving financial aid. The 95% confidence interval for p is
59 ± .07. Interpret this interval.
A) We are 95% confident that between 52% and 66% of the sampled students receive some sort of
financial aid.
B) We are 95% confident that 59% of the students are on some sort of financial aid.
C) We are 95% confident that the true proportion of all students receiving financial aid is
between .52 and .66.
D) 95% of the students receive between 52% and 66% of their tuition in financial aid.
41)
42) A local men's clothing store is being sold. The buyers are trying to estimate the percentage of items
that are outdated. They will choose a random sample from the 100,000 items in the store's
inventory in order to determine the proportion of merchandise that is outdated. The current owners
have never determined the percentage of outdated merchandise and cannot help the buyers. How
large a sample do the buyers need in order to be 98% confident that the sampling error of their
estimate is about 4%?
A) 365
B) 3394
C) 1697
D) 849
42)
6
43) A confidence interval was used to estimate the proportion of statistics students who are female. A
random sample of 72 statistics students generated the following confidence interval: (.438, .642).
Using the information above, what sample size would be necessary if we wanted to estimate the
true proportion to within 3% using 95% reliability?
A) 1110
B) 1068
C) 1061
D) 1025
43)
44) Sales of a new line of athletic footwear are crucial to the success of a company. The company
wishes to estimate the average weekly sales of the new footwear to within $200 with 95%
reliability. The initial sales indicate that the standard deviation of the weekly sales figures is
approximately $1500. How many weeks of data must be sampled for the company to get the
information it desires?
A) 15 weeks
B) 43,218 weeks
C) 111 weeks
D) 217 weeks
44)
45) In the construction of confidence intervals, if all other quantities are unchanged, an increase in the
sample size will lead to a __________ interval.
A) wider
B) biased
C) less significant
D) narrower
45)
Answer the question True or False.
46) If no estimate of p exists when determining the sample size for a confidence interval for a
proportion, we can use .5 in the formula to get a value for n.
A) True
B) False
Solve the problem.
47) How many tissues should a package of tissues contain? Researchers have determined that a person
uses an average of 50 tissues during a cold. Suppose a random sample of 10,000 people yielded the
46)
47)
following data on the number of tissues used during a cold: x = 39, s = 15. Identify the null and
alternative hypothesis for a test to determine if the mean number of tissues used during a cold is
less than 50.
A) H0 : µ > 50 vs. Ha : µ 50
B) H0 : µ = 50 vs. Ha : µ > 50
C) H0 : µ = 50 vs. Ha : µ 50
D) H0 : µ = 50 vs. Ha : µ < 50
48) A local eat-in pizza restaurant wants to investigate the possibility of starting to deliver pizzas. The
owner of the store has determined that home delivery will be successful only if the average time
spent on a delivery does not exceed 40 minutes. The owner has randomly selected 22 customers
and delivered pizzas to their homes. What hypotheses should the owner test to demonstrate that
the pizza delivery will not be successful?
A) H0 : µ = 40 vs. Ha : µ > 40
B) H0 : µ = 40 vs. Ha : µ < 40
48)
49) An insurance company sets up a statistical test with a null hypothesis that the average time for
processing a claim is 4 days, and an alternative hypothesis that the average time for processing a
claim is greater than 4 days. After completing the statistical test, it is concluded that the average
time exceeds 4 days. However, it is eventually learned that the mean process time is really 4 days.
What type of error occurred in the statistical test?
A) No error occurred in the statistical sense.
B) Type II error
C) Type III error
D) Type I error
49)
C) H0 : µ < 40 vs. Ha : µ = 40
D) H0 : µ = 40 vs. Ha : µ 40
7
50) I want to test H0 : p = .4 vs. Ha : p .4 using a test of hypothesis. If I concluded that p is .4 when, in
50)
51) A national organization has been working with utilities throughout the nation to find sites for large
wind machines that generate electricity. Wind speeds must average more than 19 miles per hour
(mph) for a site to be acceptable. Recently, the organization conducted wind speed tests at a
particular site. Based on a sample of n = 143 wind speed recordings (taken at random intervals), the
51)
fact, the true value of p is not .4, then I have made a __________.
A) Type I error
B) Type II error
C) correct decision
D) Type I and Type II error
wind speed at the site averaged x = 18.5 mph, with a standard deviation of s = 2.5 mph. To
determine whether the site meets the organization's requirements, consider the test,
H0 : µ = 19 vs. Ha : µ > 19, where µ is the true mean wind speed at the site and = .05. Fill in the
blanks. "A Type I error in the context of this problem is to conclude that the true mean wind speed
at the site _____ 19 mph when it actually _____ 19 mph."
A) exceeds; equals
B) equals; exceeds
C) equals; equals
D) exceeds; exceeds
52) A __________ is a numerical quantity computed from the data of a sample and is used in reaching a
decision on whether or not to reject the null hypothesis.
A) significance level
B) critical value
C) parameter
D) test statistic
52)
53) How many tissues should a package of tissues contain? Researchers have determined that a person
uses an average of 49 tissues during a cold. Suppose a random sample of 10,000 people yielded the
53)
following data on the number of tissues used during a cold: x = 37, s = 18. Using the sample
information provided, calculate the value of the test statistic for the relevant hypothesis test.
37 - 49
37 - 49
37 - 49
37 - 49
A) z =
B) z =
C) z =
D) z =
18
18
18
182
2
10,000
10,000
10,000
54) How many tissues should a package of tissues contain? Researchers have determined that a person
uses an average of 69 tissues during a cold. Suppose a random sample of 10,000 people yielded the
54)
following data on the number of tissues used during a cold: x = 54, s = 18. We want to test the
alternative hypothesis Ha: µ < 69. State the correct rejection region for = .05.
A) Reject H0 if z > 1.96 or z < -1.96.
B) Reject H0 if z > 1.645.
D) Reject H0 if z < -1.645.
C) Reject H0 if z < -1.96.
55) Suppose we wish to test H0 : µ = 29 vs. Ha : µ < 29. Which of the following possible sample results
gives the most evidence to support Ha (i.e., reject H0)?
A) x = 26, s = 5
B) x = 27, s = 3
C) x = 25, s = 2
8
D) x = 25, s = 5
55)
56) A national organization has been working with utilities throughout the nation to find sites for large
wind machines that generate electricity. Wind speeds must average more than 23 miles per hour
(mph) for a site to be acceptable. Recently, the organization conducted wind speed tests at a
particular site. Based on a sample of n = 42 wind speed recordings (taken at random intervals), the
56)
wind speed at the site averaged x = 23.9 mph, with a standard deviation of s = 4.4 mph. To
determine whether the site meets the organization's requirements, consider the test,
H0 : µ = 23 vs. Ha : µ > 23, where µ is the true mean wind speed at the site and = .01. Suppose the
value of the test statistic were computed to be 1.33. State the conclusion.
A) At = .01, there is insufficient evidence to conclude the true mean wind speed at the site
exceeds 23 mph.
B) We are 99% confident that the site does not meet the organization's requirements.
C) We are 99% confident that the site meets the organization's requirements.
D) At = .01, there is sufficient evidence to conclude the true mean wind speed at the site
exceeds 23 mph.
Use a hypothesis test to test the given claim.
57) A test of sobriety involves measuring the subject's motor skills. Twenty randomly selected sober
subjects take the test and produce a mean score of 41.0 with a standard deviation of 3.4. At a
significance level of 0.01, test the claim that the true mean score for all sober subjects is equal to
39.0.
A) Reject the null hypothesis of µ = 39.0 with a P-value of 0.00823. There is strong evidence that
the true mean score for sober subjects is not 39.0.
B) There is not enough information to perform the test.
C) Reject the null hypothesis of µ = 39.0 with a P-value of 0.01647. There is sufficient evidence
that the true mean score for sober subjects is different from 39.0.
D) Fail to reject the null hypothesis of µ = 39.0 with a P-value of 0.01647. There is not sufficient
evidence that the true mean score for sober subjects is different from 39.0.
E) Fail to reject the null hypothesis of µ = 39.0 with a P-value of 0.02647. There is not sufficient
evidence that the true mean score for sober subjects is different from 39.0.
For the given hypothesis test, explain the meaning of a Type I error or a Type II error, as specified.
58) In the past, the mean battery life for a certain type of flashlight battery has been 8.1 hours. The
manufacturer has introduced a change in the production method and wants to perform a
hypothesis test to determine whether the mean battery life has increased as a result. The hypotheses
are:
H : µ = 8.1 hours
0
H : µ > 8.1 hours
A
Explain the result of a Type I error.
A) The manufacturer will decide the mean battery life is less than 8.1 hours when in fact it is
greater than 8.1 hours.
B) The manufacturer will decide the mean battery life is greater than 8.1 hours when in fact it is
8.1 hours.
C) The manufacturer will decide the mean battery life is greater than 8.1 hours when in fact it is
less than 8.1 hours.
D) The manufacturer will decide the mean battery life is greater than 8.1 hours when in fact it is
greater than 8.1 hours.
E) The manufacturer will decide the mean battery life is 8.1 hours when in fact it is greater than
8.1 hours.
9
57)
58)
59) In the past, the mean battery life for a certain type of flashlight battery has been 9.4 hours. The
manufacturer has introduced a change in the production method and wants to perform a
hypothesis test to determine whether the mean battery life has increased as a result. The hypotheses
are:
H : µ = 9.4 hours
0
H : µ > 9.4 hours
A
Explain the result of a Type II error.
A) The manufacturer will decide the mean battery life is 9.4 hours when in fact it is greater than
9.4 hours.
B) The manufacturer will decide the mean battery life is less than 9.4 hours when in fact it is
greater than 9.4 hours.
C) The manufacturer will decide the mean battery life is greater than 9.4 hours when in fact it is
9.4 hours.
D) The manufacturer will decide the mean battery life is greater than 9.4 hours when in fact it is
greater than 9.4 hours.
E) The manufacturer will decide the mean battery life is 9.4 hours when in fact it is 9.4 hours.
Solve the problem.
60) If a hypothesis test were conducted using
the null hypothesis to be rejected.
A) 0.060
B) 0.020
= 0.01, to which of the following p-values would cause
C) 0.015
59)
60)
D) 0.000
61) A local eat-in pizza restaurant wants to investigate the possibility of starting to deliver pizzas. The
owner of the store has determined that home delivery will be successful only if the average time
spent on a delivery does not exceed 40 minutes. The owner has randomly selected 18 customers
and delivered pizzas to their homes in order to test whether the mean delivery time actually
exceeds 40 minutes. What assumption is necessary for this test to be valid?
A) The population variance must equal the population mean.
B) The population of delivery times must have a normal distribution.
C) None. The Central Limit Theorem makes any assumptions unnecessary.
D) The sample mean delivery time must equal the population mean delivery time.
61)
62) A local eat-in pizza restaurant wants to investigate the possibility of starting to deliver pizzas. The
owner of the store has determined that home delivery will be successful only if the average time
spent on a delivery does not exceed 31 minutes. The owner has randomly selected 15 customers
and delivered pizzas to their homes in order to test whether the mean delivery time actually
exceeds 31 minutes. Suppose the p-value for the test was found to be .0265. State the correct
conclusion.
A) At = .05, we fail to reject H0 .
B) At = .025, we fail to reject H0 .
62)
63) An industrial supplier has shipped a truckload of teflon lubricant cartridges to an aerospace
customer. The customer has been assured that the mean weight of these cartridges is in excess of
the 10 ounces printed on each cartridge. To check this claim, a sample of n = 10 cartridges are
randomly selected from the shipment and carefully weighed. Summary statistics for the sample
63)
C) At
D) At
= .03, we fail to reject H0 .
= .02, we reject H0 .
are: x = 10.11 ounces, s = .30 ounce. To determine whether the supplier's claim is true, consider the
test, H0: µ = 10 vs. Ha: µ > 10, where µ is the true mean weight of the cartridges. Find the rejection
region for the test using = .01.
A) t > 3.25, where t depends on 9 df
C) t > 2.821, where t depends on 9 df
B) z > 2.33
D) |z| > 2.58
10
64) A bottling company produces bottles that hold 8 ounces of liquid. Periodically, the company gets
complaints that their bottles are not holding enough liquid. To test this claim, the bottling company
randomly samples 17 bottles and finds the average amount of liquid held by the bottles is 7.6
ounces with a standard deviation of .3 ounce. Which of the following is the set of hypotheses the
company wishes to test?
A) H0 : µ < 8 vs. Ha: µ = 8
B) H0 : µ = 8 vs. Ha: µ < 8
64)
65) A bottling company produces bottles that hold 12 ounces of liquid. Periodically, the company gets
complaints that their bottles are not holding enough liquid. To test this claim, the bottling company
randomly samples 20 bottles and finds the average amount of liquid held by the bottles is 11.6
ounces with a standard deviation of .3 ounce. Calculate the appropriate test statistic.
A) t = -26.667
B) t = -5.812
C) t = -3.266
D) t = -5.963
65)
66) The business college computing center wants to determine the proportion of business students who
have laptop computers. If the proportion exceeds 25%, then the lab will scale back a proposed
enlargement of its facilities. Suppose 200 business students were randomly sampled and 65 have
laptops. Find the rejection region for the corresponding test using = .01.
A) Reject H0 if z > 2.33.
B) Reject H0 if z < -2.33.
C) Reject H0 if z > 2.575 or z < -2.575.
D) Reject H0 if z = 2.33.
66)
67) The business college computing center wants to determine the proportion of business students who
have laptop computers. If the proportion differs from 35%, then the lab will modify a proposed
enlargement of its facilities. Suppose a hypothesis test is conducted and the test statistic is 2.6. Find
the p-value for a two-tailed test of hypothesis.
A) .0047
B) .4906
C) .0094
D) .4953
67)
68) The business college computing center wants to determine the proportion of business students who
have laptop computers. If the proportion exceeds 30%, then the lab will scale back a proposed
enlargement of its facilities. Suppose 200 business students were randomly sampled and 65 have
laptops. What assumptions are necessary for this test to be satisfied?
A) The sample size n satisfies both np0 15 and nq0 5.
B) The sample size n satisfies n 30.
C) The population has an approximately normal distribution.
D) The sample proportion is close to .5.
68)
69) A company claims that 9 out of 10 doctors (i.e., 90%) recommend its brand of cough syrup to their
patients. To test this claim against the alternative that the actual proportion is less than 90%, a
random sample of doctors was taken. Suppose the test statistic is z = -1.95. Can we conclude that
H0 should be rejected at the a) = .10, b) = .05, and c) = .01 level?
69)
C) H0 : µ = 8 vs. Ha: µ 8
D) H0 : µ = 8 vs. Ha: µ > 8
A) a) yes; b) yes; c) no
C) a) no; b) no; c) no
B) a) yes; b) yes; c) yes
D) a) no; b) no; c) yes
11
70) A company reports that 80% of its employees participate in the company’s stock purchase plan. A
random sample of 50 employees was asked the question, "Do you participate in the stock purchase
plan?" The answers are shown below.
yes
no
no
yes
no
no
yes
yes
no
yes
no
yes
yes
no
yes
yes
yes
no
yes
no
no
no
yes
yes
yes
no
yes
yes
yes
yes
yes
no
no
yes
yes
yes
no
yes
yes
yes
no
yes
yes
no
yes
70)
no
yes
yes
yes
yes
Perform the appropriate test of hypothesis to investigate your suspicion that fewer than 80% of the
company's employees participate in the plan. Use = .05.
^ 32
.64 - .8
= .64; The test statistic is z =
-2.828. The rejection region is z < 1.645.
A) p =
50
(.8)(.2) / 50
B)
Since the test statistic falls within the rejection region, we reject the null hypothesis in favor of
the alternative hypothesis and conclude that fewer than 80% of the company’s employees
participate in the stock plan.
71) The following random sample was selected from a normal population: 9, 11, 8, 10, 14, 8. Construct a
95% confidence interval for the population mean µ.
s
2.28
= 10 ± 2.571
= 10 ± 2.393
A) x = 10; s = 2.28; x ± t /2
n
6
71)
B)
72) Suppose you wanted to estimate a binomial proportion, p, correct to within .01 with probability 0.
99. What size sample would need to be selected if p is known to be approximately 0.85?
z /2 2
p(1 - p).
A) To determine the sample size necessary to estimate p, we use n =
SE
72)
For confidence coefficient .99, 1 - = .99
= 1 - .99 = .01.
/2 = .01/2 = .005.
z /2 = z .005 = 2.575.
2.575 2
n=
(.85)(1 - .85) = 8454.04688. Round up to n = 8455.
.01
B)
73) State University uses thousands of fluorescent light bulbs each year. The brand of bulb it currently
uses has a mean life of 910 hours. A competitor claims that its bulbs, which cost the same as the
brand the university currently uses, have a mean life of more than 910 hours. The university has
decided to purchase the new brand if, when tested, the evidence supports the manufacturer's claim
at the .10 significance level. Suppose 75 bulbs were tested with the following results: x = 936 hours,
s = 95 hours. Find the rejection region for the test of interest to the State University.
A) To determine if the mean exceeds 910 hours, we test:
H0: µ = 910 vs. Ha : µ > 910
B)
The rejection region requires = .10 in the upper tail of the z distribution. From a z table, we
find z.10 = 1.28. The rejection region is z > 1.28.
12
73)
74) A supermarket sells rotisserie chicken at a fixed price per chicken rather than by the weight of the
chicken. The store advertises that the average weight of their chickens is 4.6 pounds. A random
sample of 30 of the store's chickens yielded the weights (in pounds) shown below.
74)
4.4 4.7 4.6 4.4 4.5 4.3 4.6 4.5 4.6 4.9
4.6 4.8 4.3 4.4 4.7 4.5 4.2 4.3 4.1 4.0
4.5 4.6 4.2 4.4 4.7 4.8 5.0 4.2 4.1 4.5
Test whether the population mean weight of the chickens is less than 4.6 pounds. Use = .05.
4.48 - 4.6
-2.677; rejection region: z <
A) H0 : µ = 4.6; Ha : µ < 4.6; x = 4.48; s = .2455; z =
.2455 / 30
B)
-1.645;
Since the test statistic falls within the rejection region, we reject the null hypothesis in favor of
the alternative hypothesis. There is evidence that the mean weight of the chicken is less than
4.6 pounds.
75) A random sample of n = 18 observations is selected from a normal population to test H0: µ = 145
against Ha : µ
145 at
75)
= .10. Specify the rejection region.
A) Using 17 degrees of freedom, t.10/2 = t.05 = 1.740. The rejection region is t < -1.740 or
B)
t > 1.740.
76) The hypotheses for H0 : µ = 65 and Ha: µ > 65 are testedat
= .05 based on a sample of the size n =
76)
250 . Sketch the appropriate rejection region.
A)
B)
77) A random sample of n = 12 observations is selected from a normal population to test H0: µ = 22.1
against Ha : µ > 22.1 at
= .05. Specify the rejection region.
A) Using 11 degrees of freedom, t.05 = 1.796. The rejection region is t > 1.796.
B)
13
77)
78) A random sample of n = 15 observations is selected from a normal population to test H0: µ = 2.89
against Ha : µ < 2.89 at
78)
= .01. Specify the rejection region.
A) Using 14 degrees of freedom, t.01 = 2.624. The rejection region is t < -2.624.
B)
79) In a test of H0 : µ = 65 against Ha: µ > 65, the large sample data yielded the test statistic z = 1.38.
79)
80) In a test of H0 : µ = 70 against Ha: µ 70, the larege sample data yielded the test statistic z = 2.11.
80)
81) A sample of 6 measurements, randomly selected from a normally distributed population, resulted
81)
Find and interpret the p-value for the test.
A) p-value = P(z >1.38) = .5 - .4162 = .0838; The probability of a test statistic even more
contradictory to the null hypothesis than the one observed is .0838.
B)
Find and interpret the p-value for the test.
A) p-value = P(z < -2.11 or z > 2.11) = 2(.5 - .4826) = .0348; The probability of a test statistic even
more contradictory to the null hypothesis than the one observed is .0348.
B)
in the following summary statistics: x = 9.1, s = 1.5. Test the null hypothesis that the mean of the
population is 10 against the alternative hypothesis µ < 10. Use = .05.
9.1 - 10
-1.47. The rejection region is t < -2.015. Since the test statistic
A) The test statistic is t =
1.5 / 6
B)
does not fall in the rejection region, we can not reject the null hypothesis in favor of the
alternative hypothesis. We cannot conclude that the true population mean is actually less than
10.
82) A sample of 8 measurements, randomly selected from a normally distributed population, resulted
82)
in the following summary statistics: x = 5.2, s = 1.1. Test the null hypothesis that the mean of the
population is 4 against the alternative hypothesis µ 4. Use = .05.
5.2 - 4
3.09. The rejection region is t < -2.365 or t > 2.365. Since the
A) The test statistic is t =
1.1 / 8
B)
test statistic falls in the rejection region, we reject the null hypothesis in favor of the
alternative hypothesis. We conclude that 4 is not the true population mean.
83) A random sample of 8 observations from an approximately normal distribution is shown below.
5
6
4
5
8
6
5
3
Find the observed level of significance for the test of H0 : µ = 5 against Ha: µ
5. Interpret the result.
A) p = .649; The probability of a test statistic even more contradictory than the one observed is
.649.
B)
14
83)
84) Increasing numbers of businesses are offering child-care benefits for their workers. However, one
union claims that more than 85% of firms in the manufacturing sector still do not offer any
child-care benefits. A random sample of 420 manufacturing firms is selected, and only 26 of them
offer child-care benefits. Specify the rejection region that the union will use when testing at = .10.
A) To determine if more than 85% of the firms do not offer any child-care benefits, we test:
84)
H0: p = .85 vs. Ha : p > .85
B)
The rejection region requires
is z > z.10 = 1.28.
= .10 in the upper tail of the z distribution. The rejection region
85) Increasing numbers of businesses are offering child-care benefits for their workers. However, one
union claims that more than 85% of firms in the manufacturing sector still do not offer any
child-care benefits. A random sample of 270 manufacturing firms is selected and asked if they offer
child-care benefits. Suppose the p-value for this test was reported to be p = .1192. State the
conclusion of interest to the union. Use = .10.
A) At = .10, < p-value = .1192, so H0 cannot be rejected. There is insufficient evidence to
B)
indicate that more than 85% of the firms do not offer any child-care benefits.
15
85)
STT 315 Practice Problems III for Sections 5.1 - 7.6 ANSWERS
For your convenience open-end questions 70 - 85 were converted to bimodal so you can see
solutions.
1)
2)
3)
4)
5)
6)
7)
8)
A
B
A
B
A
A
A
9) a.
b.
10)
11)
12)
13)
14)
15)
c.
B
A
A
D
C
μ = E(x) = .1(3) + .7(5) + .2(7) = 5.2
E( ) = .01(3) + .14(4) + .53(5) + .28(6) + .04(7) = 5.2; Since E( ) = μ,
= μ = 65;
16)
17)
18)
19)
20)
21)
22)
23)
24)
25)
26)
27)
28)
29)
30)
31)
32)
33)
34)
35)
36)
37)
38)
D
A
C
D
A
D
A
A
D
C
C
C
B
A
C
D
B
D
D
C
B
B
A
=
=
= 1.5
is an unbiased estimator of
μ.
39)
40)
41)
42)
43)
44)
45)
46)
47)
48)
49)
50)
51)
52)
53)
54)
55)
56)
57)
58)
59)
60)
61)
62)
63)
64)
65)
66)
67)
68)
69)
70)
D
A
C
D
C
D
D
A
D
A
D
B
A
D
D
D
C
A
D
B
A
D
B
B
C
B
D
A
C
A
A
See problem. For your convenience open-end questions 70 - 85 were converted to bimodal so
you can see solutions.