time is less than 15 minutes.
2. At a certain manufacturing plant, a machine produced ball bearings that should have a diameter of 0.50
mm. If the machine produces ball bearings that are either too small or too large, the ball bearings must
be scrapped. Every hour, a quality control manager takes a random sample of 30 ball bearings to test to
see if the process is “out of control” (i.e. to test to see if the average diameter differs from 0.50 mm).
a.
State the hypotheses associated with the manager’s test.
b.
Describe a Type I error for this situation.
c.
Describe a Type II error for this situation.
control.
out of control.
3. A fast food restaurant is considering a promotion that will offer customers to purchase a toy featuring a
cartoon movie character. If more than 20% of the customers purchase the toy, the promotion will be
profitable. A sample of 50 restaurants is used to test the promotion.
a.
State the hypotheses associated with the restaurant’s test.
b.
Describe a Type I error for this situation.
c.
Describe a Type II error for this situation.
4. The average gasoline price of one of the major oil companies has been $1.00 per gallon. Because of
shortages in production of crude oil, it is believed that there has been a significant increase in the
average price. In order to test this belief, we randomly selected a sample of 36 of the company’s gas
stations and determined that the average price for the stations in the sample was $1.10. Assume that
the standard deviation of the population () is $0.12.
a.
State the null and the alternative hypotheses.
b.
Test the claim at = .05.
c.
What is the p-value associated with the above sample results?
been an increase in the average price
5. “D” size batteries produced by MNM Corporation have had a life expectancy of 87 hours. Because of
an improved production process, the company believes that there has been an increase in the life
expectancy of its “D” size batteries. A sample of 36 batteries showed an average life of 88.5 hours.
Assume from past information that it is known that the standard deviation of the population is 9 hours.
a.
Use a 0.01 level of significance, and test to determine if there has been an increase in the life
expectancy of the “D” size batteries.
b.
What is the p-value associated with the sample results? What is your conclusion, based on the
p-value?
6. At a local university, a sample of 49 evening students was selected in order to determine whether the
average age of the evening students is significantly different from 21. The average age of the students
in the sample was 23 years. The population standard deviation is known to be 3.5 years. Determine
whether or not the average age of the evening students is significantly different from 21. Use a 0.1
level of significance.
7. In order to determine the average price of hotel rooms in Atlanta, a sample of 64 hotels was selected. It
was determined that the average price of the rooms in the sample was $112. The population standard
deviation is known to be $16. Use a 0.05 level of significance and determine whether or not the
average room price is significantly different from $108.50.
8. A sample of 81 account balances of a credit company showed an average balance of $1,200. The
population standard deviation is $126. You want to determine if the mean of all account balances is
significantly different from $1,150. Use a .05 level of significance.
9. A lathe is set to cut bars of steel into lengths of 6 centimeters. The lathe is considered to be in perfect
adjustment if the average length of the bars it cuts is 6 centimeters. A sample of 121 bars is selected
randomly and measured. It is determined that the average length of the bars in the sample is 6.08
centimeters. The population standard deviation is 0.44 centimeters. Determine whether or not the lathe
is in perfect adjustment. Use a .05 level of significance.
10. Bastien, Inc. has been manufacturing small automobiles that have averaged 50 miles per gallon of
gasoline in highway driving. The company has developed a more efficient engine for its small cars and
now advertises that its new small cars average more than 50 miles per gallon in highway driving. An
independent testing service road-tested 36 of the automobiles. The sample showed an average of 51.5
miles per gallon. The population standard deviation is 6 miles per gallon.
a.
With a 0.05 level of significance, test to determine whether or not the manufacturer’s
advertising campaign is legitimate.
b.
What is the p-value associated with the sample results?
b.
0.0668
11. A carpet company advertises that it will deliver your carpet within 15 days of purchase. A sample of
49 past customers is taken. The average delivery time in the sample was 16.2 days. Assume the
population standard deviation is known to be 5.6 days.
a.
State the null and alternative hypotheses.
b.
Using a critical value, test the null hypothesis at the 5% level of significance.
c.
Using a p-value, test the hypothesis at the 5% level of significance.
d.
What type of error may have been committed for this hypothesis test?
May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Do not reject H0, 1.5 1.645, there is not sufficient evidence at = .05 to conclude that the
Do not reject H0, 0.0668 0.05 (same conclusion as part b)
d.
A Type II error may have been committed since we did not reject H0
12. A student believes that the average grade on the statistics final examination is 87. A sample of 36 final
examinations is taken. The average grade in the sample is 83.96. The population variance is 144.
a.
State the null and alternative hypotheses.
b.
Using a critical value, test the hypothesis at the 5% level of significance.
c.
Using a p-value, test the hypothesis at the 5% level of significance.
d.
Using a confidence interval, test the hypothesis at the 5% level of significance.
e.
Compute the probability of a Type II error if the average grade on the final is 85.
Do no reject H0, -1.96 -1.52 1.96, there is not sufficient evidence at = .05 to conclude
Do not reject H0, 0.1286 0.05 (same conclusion as part b)
d.
80.04 to 87.88; do not reject H0 (same conclusion as part b)
e.
0.8315
13. A carpet company advertises that it will deliver your carpet within 15 days of purchase. A sample of
49 past customers is taken. The average delivery time in the sample was 16.2 days. The population
standard deviation is 5.6 days.
a.
State the null and alternative hypotheses.
b.
Using a critical value, test the null hypothesis at the 5% level of significance.
c.
Using a p-value, test the hypothesis at the 5% level of significance.
d.
Compute the probability of a Type II error if the true average delivery time is 17 days after
purchase.
Do not reject H0, 1.5 1.645
Do not reject H0, 0.0668 0.05
d.
0.1949
14. The sponsors of televisions shows targeted at the market of 5 – 8 year olds want to test the hypothesis
that children watch television at most 20 hours per week. The population of viewing hours per week is
known to be normally distributed with a standard deviation of 6 hours. A market research firm
conducted a random sample of 30 children in this age group. The resulting data follows:
19.5
17.5
19.4
18.4
14.6
12.5
19.1
30.9
22.2
11.8
27.7
25.3
27.4
16.1
20.6
32.9
27.0
17.1
20.1
17.7
At a .10 level of significance, use Excel to test the sponsors’ hypothesis.
=COUNT(A2:A31)
30
=AVERAGE(A2:A31)
20.266667
6
6
20
20
=D4/SQRT(D1)
1.095445
=(D2-D5)/D7
0.243432
=NORM.S.DIST(D8,TRUE)
0.596165
0.403835
=2*(MIN(D10,D11))
0.807670
PTS: 1
15. At a certain manufacturing plant, a machine produces ball bearings that should have a diameter of
0.500 mm. If the machine produces ball bearings that are either too small or too large, the ball bearings
must be scrapped. Every hour, a quality control manager takes a random sample of 36 ball bearings to
test to see if the process is “out of control” (i.e. to test to see if the average diameter differs from 0.500
mm). Assume that the process is maintaining the desired standard deviation of .06 mm. The results
from the latest sample follow:
0.468
0.421
0.448
0.346
0.452
0.465
0.558
0.526
0.354
0.447
0.411
0.453
0.456
0.529
0.570
0.319
0.471
0.499
0.405
0.557
0.468
0.421
0.448
0.346
At a .01 level of significance, use Excel to test whether the process is “out of control.”
16. From a population of cans of coffee marked “12 ounces,” a sample of 25 cans is selected and the
contents of each can are weighed. The sample revealed a mean of 11.8 ounces and a standard deviation
of 0.5 ounces. Test to see if the mean of the population is at least 12 ounces. (Assume the population is
normally distributed.) Use a .05 level of significance.
17. In the past the average age of employees of a large corporation has been 40 years. Recently, the
company has been hiring older individuals. In order to determine whether there has been an increase in
the average age of all the employees, a sample of 25 employees was selected. The average age in the
sample was 45 years with a standard deviation of 5 years. Assume the distribution of the population is
normal. Let = .05.
a.
State the null and the alternative hypotheses.
b.
Test to determine whether or not the mean age of all employees is significantly more than 40
years.
of all employees of the large corporation has increased
=AVERAGE(A2:A37)
0.457278
0.500
0.500
=(D2-D5)/D7
–4.272223
0.99999
18. A soft drink filling machine, when in perfect adjustment, fills the bottles with 12 ounces of soft drink.
A random sample of 25 bottles is selected, and the contents are measured. The sample yielded a mean
content of 11.88 ounces, with a standard deviation of 0.24 ounces. With a 0.05 level of significance,
test to see if the machine is in perfect adjustment. Assume the distribution of the population is normal.
19. A sample of 16 cookies is taken to test the claim that each cookie contains at least 9 chocolate chips.
The average number of chocolate chips per cookie in the sample was 7.875 with a standard deviation
of 1. Assume the distribution of the population is normal.
a.
State the null and alternative hypotheses.
b.
Using a critical value, test the hypothesis at the 1% level of significance.
c.
Using a p-value, test the hypothesis at the 1% level of significance.
d.
Compute the probability of a Type II error if the true number of chocolate chips per cookie is 8.
Reject H0, -4.5 -2.602
c.
Reject H0; the p-value is less than 0.005
d.
A Type II error has not been committed since H0 was rejected.
20. Nancy believes that the average running time of movies is equal to 140 minutes. A sample of 4 movies
was taken and the following running times were obtained. Assume the distribution of the population is
normally distributed.
150
150
180
170
a.
State the null and alternative hypotheses.
b.
Using a critical value, test the hypothesis at the 10% level of significance.
c.
Using a p-value, test the hypothesis at the 10% level of significance.
d.
Using a confidence interval, test the hypothesis at the 10% level of significance.
e.
Could a Type II error have been committed in this hypothesis test?
running time of movies differs from 140 minutes
b)
d.
144.85 to 180.15; Reject H0 (same conclusion as part b)
e.
A Type II error could not have been committed since H0 was rejected.
21. You are given the following information obtained from a random sample of 5 observations.
20
18
17
22
18
At a 10% level of significance, use Excel to determine whether or not the mean of the population from
which this sample was taken is significantly less than 21. (Assume the population is normally
distributed.)
=STDEV.S(A2:A6)
2
21
21
=(D2-D5)/D7
–2.236068
=T.DIST(D8,D9,TRUE)
0.04450
22. You are given the following information obtained from a random sample of 4 observations.
25
47
32
56
At a .05 level of significance, use Excel to determine whether or not the mean of the population from
which this sample was taken is significantly different from 48. (Assume the population is normally
distributed.)
=COUNT(A2:A5)
4
=STDEV.S(A2:A5)
14.07125
=D3/SQRT(D1)
7.03562
=(D2-D5)/D7
–1.13707
=D1-1
3
=T.DIST(D8,D9,TRUE)
0.16906
0.83094
=2*MIN(D11,D12)
0.33811
23. A group of young businesswomen wish to open a high fashion boutique in a vacant store, but only if
the average income of households in the area is more than $45,000. A random sample of 9 households
showed the following results.
$48,000
$44,000
$46,000
$43,000
$47,000
$46,000
$44,000
$42,000
$45,000
Use the statistical techniques in Excel to advise the group on whether or not they should locate the
boutique in this store. Use a .05 level of significance. (Assume the population is normally distributed.)
=COUNT(A2:A10)
9
=AVERAGE(A2:A10)
45000
=STDEV.S(A2:A10)
1936.49167
45000
45000
=D3/SQRT(D1)
645.49722
=(D2-D5)/D7
0
=D1-1
8
=T.DIST(D8,D9,TRUE)
0.50000
0.50000
=2*MIN(D11,D12)
1.00000
24. In a television commercial, the manufacturer of a toothpaste claims that at least 4 out of 5 dentists
recommend its product. A consumer-protection group wants to test that claim. Identify the hypotheses.
25. A manufacturer is considering a new production method. The current method produces 94%
non-defective (good) parts. The new method will be implemented if it produces more non-defectives
than the current method. Identify the hypotheses.
26. Consider the following hypothesis test:
Ho: p = 0.5
Ha: p 0.5
A sample of 800 provided a sample proportion of 0.58.
a.
Using = 0.05, what is the rejection rule?
b.
Determine the standard error of the proportion.
c.
Compute the value of the test statistic z. What is your conclusion?
d.
Determine the p-value.
a.
Reject H0 if z –1.96 or if z 1.96
b.
0.01767
is significantly different from 0.5
d.
zero
27. A new soft drink is being market tested. A sample of 400 individuals participated in the taste test and
80 indicated they like the taste.
a.
At a 5% significance level, test to determine if at least 22% of the population will like the new
soft drink.
b.
Determine the p-value.
a.
5% to conclude that fewer than 22% of the population like the new soft drink
b.
0.1587
28. A student believes that no more than 20% (i.e., 20%) of the students who finish a statistics course get
an A. A random sample of 100 students was taken. Twenty-four percent of the students in the sample
received A’s.
a.
State the null and alternative hypotheses.
b.
Using a critical value, test the hypothesis at the 1% level of significance.
c.
Using a p-value, test the hypothesis at the 1% level of significance.
than 20% of the students get an A.
Do not reject H0; 0.1587 0.01 (same conclusion as part b)
29. For each shipment of parts a manufacturer wants to accept only those shipments with at most 10%
defective parts. A large shipment has just arrived. A quality control manager randomly selects 50 of
the parts from the shipment and finds that 6 parts are defective. Is this sufficient evidence to reject the
entire shipment? Use a .05 level of significance to conduct the appropriate hypothesis test.
30. A national poll reported that 58% of those with Internet access have made purchases online. To
investigate whether this percentage applies to its own state, a legislator commissions a study. A
random sample of 400 state residents who have Internet access is taken. Of those 400 respondents, 215
said that they have made purchases online. Does this sample provide sufficient evidence to conclude
that the state differs from the nation with respect to making purchases online? Use the p-value to
conduct the hypothesis test and use a .05 level of significance.
31. An official of a large national union claims that the fraction of women in the union is not significantly
different from one-half. Using the sample information reported below, carry out a test of this
statement. Use a .05 level of significance.
sample size
400
women
168
men
232
32. A manufacturer claims that at least 40% of its customers use coupons. A study of 25 customers is
undertaken to test that claim. The results of the study follow.
yes
no
yes
no
no
yes
no
no
yes
no
no
yes
no
yes
yes
At a .05 level of significance, use Excel to test the manufacturer’s claim.
=COUNTA(A2:A26)
25
Yes
yes
=COUNTIF(A2:A26,D2)
9
=D3/D1
0.36
0.4
0.4
=SQRT(D4*(1-D4)/D1)
0.096
=(D4-D6)/D8
–0.41666667
=NORM.S.DIST(D9,TRUE)
0.338461157
0.661538843
=2*MIN(D11,D12)
0.67692
33. Several years ago the proportion of Americans aged 18 – 24 who invested in the stock market was
0.20. A random sample of 25 Americans in this age group was recently taken. They were asked
whether or not they invested in the stock market. The results follow:
yes
no
no
yes
no
no
yes
no
no
yes
no
no
no
no
no
no
no
no
yes
no
no
no
yes
no
no
At a .05 level of significance, use Excel to determine whether or not the proportion of Americans 18 –
24 years old that invest in the stock market has changed.
yes
yes
=COUNTIF(A2:A26,D2)
6
0.2
0.2
=SQRT(D6*(1-D6)/D1)
0.08
=(D4-D6)/D8
0.50
=NORM.S.DIST(D9,TRUE)
0.6914625
0.3085375
34. Identify the null and alternative hypotheses for the following problems.
a.
The manager of a restaurant believes that it takes a customer no more than 25 minutes to eat
lunch.
b.
Economists have stated that the marginal propensity to consume is at least 90¢ out of every
dollar.
c.
It has been stated that 75 out of every 100 people who go to the movies on Saturday night buy
popcorn.
35. Fast ‘n Clean operates 12 laundromats on the east side of the city. All of Fast ‘n Clean’s clothes
dryers have a label stating “20 minutes for $1.00.” You question the accuracy of the dryers’ clocks
and decide to conduct an observational study. You randomly select 36 dryers in several different Fast
‘n Clean locations, put $1.00 in each and time the drying cycle. The sample mean drying time is 20
minutes and 25 seconds. The manufacturer of the dryer states that the standard deviation for
20-minute drying cycles is 1 minute.
a. Using the sample data and
= .05, test the validity of the label on the dryers. Apply the p-value
and critical value approaches to conducting the two-tail hypothesis test.
b. Conduct the same two-tail hypothesis test, but this time use the confidence interval approach to
hypothesis testing.
36. The board of directors of a corporation has agreed to allow the human resources manager to move to
the next step in planning day care service for employees’ children if the manager can prove that at least
25% of the employees have interest in using the service. The HR manager polls 300 employees and
90 say they would seriously consider utilizing the service. At the
= .10 level of significance, is
there enough interest in the service to move to the next planning step?