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.
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?
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?
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?
EMBS4 TB09 – 17
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.
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.
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
29.7
17.5
10.4
19.4
18.4
14.6
10.1
12.5
18.2
19.1
30.9
22.2
19.8
11.8
19.0
27.7
25.3
27.4
26.5
16.1
21.7
20.6
32.9
27.0
15.6
17.1
19.2
20.1
17.7
At a .10 level of significance, use Excel to test the sponsors’ hypothesis.
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
0.521
0.421
0.476
0.448
0.346
0.452
0.513
0.465
0.395
0.558
0.526
0.354
0.474
0.447
0.405
0.411
0.453
0.456
0.477
0.529
0.440
0.570
0.319
0.471
0.480
0.499
0.446
0.405
0.557
0.468
0.521
0.421
0.476
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.
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.
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?
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.)
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.)
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.)
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.
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.
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.
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
no
yes
yes
no
yes
no
no
yes
no
no
no
no
yes
no
no
no
no
yes
no
no
yes
no
yes
At a .05 level of significance, use Excel to test the manufacturer’s claim.
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.
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?