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Hypothesis Testing
In a lower one-tail hypothesis test situation, the p-value is determined to be 0.2. If the sample
size for this test is 51, the t statistic has a value of
For a one-tailed test (lower tail) with 22 degrees of freedom at 95% confidence, the value of t
=
For a two-tailed test, a sample of 20 at 80% confidence, t =
For a two-tailed test at 98.4% confidence, Z =
A random sample of 100 people was taken. Eighty-five of the people in the sample favored
Candidate A. We are interested in determining whether or not the proportion of the population
in favor of Candidate A is significantly more than 80%.
The quality control engineer for a furniture manufacturer is interested in the mean amount of
force necessary to produce cracks in stressed oak furniture. She performs a two-tail test of the
null hypothesis that the mean for the stressed oak furniture is 650. The calculated value of
the Z test statistic is a positive number that leads to a p-value of 0.080 for the test.
Suppose the engineer had decided that the alternative hypothesis to test was that the mean was
less than 650. What would be the p-value of this one-tail test?
A student claims that he can correctly identify whether a person is a business major or an
agriculture major by the way the person dresses. Suppose in actuality that if someone is a
business major, he can correctly identify that person as a business major 87% of the time. When
a person is an agriculture major, the student will incorrectly identify that person as a business
major 16% of the time. Presented with one person and asked to identify the major of this person
(who is either a business or an agriculture major), he considers this to be a hypothesis test with
the null hypothesis being that the person is a business major and the alternative that the person
is
an
agriculture
major.
What is the power of the test?
A drug company is considering marketing a new local anesthetic. The effective time of the
anesthetic the drug company is currently producing has a normal distribution with an mean of
7.4 minutes with a standard deviation of 1.2 minutes. The chemistry of the new anesthetic is
such that the effective time should be normally distributed with the same standard deviation,
but the mean effective time may be lower. If it is lower, the drug company will market the new
anesthetic; otherwise, they will continue to produce the older one. A sample of size 36 results
in a sample mean of 7.1. A hypothesis test will be done to help make the decision.
The appropriate hypotheses are:
The average life expectancy of tires produced by the Whitney Tire Company has been 40,000
miles. Management believes that due to a new production process, the life expectancy of their
tires has increased. In order to test the validity of their belief, the correct set of hypotheses is
The average manufacturing work week in metropolitan Chattanooga was 40.1 hours last
year. It is believed that the recession has led to a reduction in the average work week. To test
the validity of this belief, the hypotheses are
A machine is designed to fill toothpaste tubes with 5.8 ounces of toothpaste. The manufacturer
does not want any underfilling or overfilling. The correct hypotheses to be tested are
The average hourly wage of computer programmers with 2 years of experience has been
$21.80. Because of high demand for computer programmers, it is believed there has been a
significant increase in the average wage of computer programmers. To test whether or not
there has been an increase, the correct hypotheses to be tested are
In the past, 75% of the tourists who visited Chattanooga went to see Rock City. The
management of Rock City recently undertook an extensive promotional campaign. They are
interested in determining whether the promotional campaign actually increased the proportion
of tourists visiting Rock City. The correct set of hypotheses is
Inferences About Mean & Proportions with Two Populations
Exhibit 10-11
An insurance company selected samples of clients under 18 years of age and over 18 and
recorded the number of accidents they had in the previous year. The results are shown below.
Under Age of 18
Over Age of 18
n1 = 500
n2 = 600
Number of accidents = 180 Number of accidents = 150
We are interested in determining if the accident proportions differ between the two age groups.
Refer to Exhibit 10-11. The pooled proportion is
Exhibit10-3
A statistics teacher wants to see if there is any difference in the abilities of students enrolled in
statistics today and those enrolled five years ago. A sample of final examination scores from
students enrolled today and from students enrolled five years ago was taken. You are given the
following information.
Today Five Years Ago
82
88
2
σ 112.5
54
n 45
36
Refer to Exhibit 10-3. The 95% confidence interval for the difference between the two
population means is
Exhibit10-6
The management of a department store is interested in estimating the difference between the
mean credit purchases of customers using the store's credit card versus those customers using
a national major credit card. You are given the following information.
Sample size
Sample mean
Population standard deviation
Store's Card Major Credit Card
64
49
$140
$125
$10
$8
Refer to Exhibit 10-6. At 95% confidence, the margin of error is
Exhibit10-9
Two major automobile manufacturers have produced compact cars with the same size engines.
We are interested in determining whether or not there is a significant difference in the MPG
(miles per gallon) of the two brands of automobiles. A random sample of eight cars from each
manufacturer is selected, and eight drivers are selected to drive each automobile for a specified
distance. The following data show the results of the test.
Driver Manufacturer A Manufacturer B
1
32
28
2
27
22
3
26
27
4
26
24
5
25
24
6
29
25
7
31
28
8
25
27
Refer to Exhibit 10-9. The mean for the differences is
Exhibit10-2
The
following
information
was
obtained
from
matched
samples.
The daily production rates for a sample of workers before and after a training program are
shown below.
Worker Before After
1
20
22
2
25
23
3
27
27
4
23
20
5
22
25
6
20
19
7
17
18
Refer to Exhibit 10-2. The point estimate for the difference between the means of the two
populations is
Exhibit10-1
Salary information regarding male and female employees of a large company is shown below.
Male Female
Sample Size
64
36
Sample Mean Salary (in $1,000) 44
41
Population Variance ()
128
72
Refer to Exhibit 10-1. At 95% confidence, the margin of error is
Exhibit10-3
A statistics teacher wants to see if there is any difference in the abilities of students enrolled in
statistics today and those enrolled five years ago. A sample of final examination scores from
students enrolled today and from students enrolled five years ago was taken. You are given the
following information.
Today Five Years Ago
82
88
2
σ 112.5
54
n 45
36
Refer to Exhibit 10-3. The test statistic for the difference between the two population means is
Exhibit10-10
The results of a recent poll on the preference of shoppers regarding two products are shown
below.
Product Shoppers Surveyed
A
800
B
900
Shoppers Favoring
This Product
560
612
Refer to Exhibit 10-10. The 95% confidence interval estimate for the difference between the
populations favoring the products is
Exhibit10-5
The following information was obtained from matched samples.
Individual Method 1 Method 2
1
7
5
2
5
9
3
6
8
4
7
7
5
5
6
Refer to Exhibit 10-5. The 95% confidence interval for the difference between the two
population means is
Exhibit10-13
In order to determine whether or not there is a significant difference between the hourly wages
of two companies, the following data have been accumulated.
Company 1 Company 2
n1 = 80
n2 = 60
= $10.80 = $10.00
= $2.00
= $1.50
Refer to Exhibit 10-13. The null hypothesis for this test is
Exhibit10-10
The results of a recent poll on the preference of shoppers regarding two products are shown
below.
Shoppers Favoring
Product Shoppers Surveyed This Product
A
800
560
B
900
612
Refer to Exhibit 10-10. At 95% confidence, the margin of error is
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 64 employees was selected.
The average age in the sample was 45 years with a standard deviation of 16 years. Let α = .05.
a. State the null and the alternative hypotheses.
b. Compute the test statistic.
c. Using the p-value approach, test to determine whether or not the mean age of all
employees is significantly more than 40 years.
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 at least $25,000. A random sample of 9
households showed the following results
$28,000 $24,000 $26,000 $25,000
$23,000 $27,000 $26,000 $22,000
$24,000
Assume the population of incomes is normally distributed.
a. Compute the sample mean and the standard deviation.
b. State the hypotheses for this problem.
c. Compute the test statistic.
d. At 95% confidence using the p-value approach, what is your conclusion?
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 the critical value approach, test the hypotheses at the 1% level of significance.
c. Using the p-value approach, test the hypotheses at the 1% level of significance.
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. Determine the standard error of the proportion.
b. Compute the value of the test statistic.
c. Determine the p-value, and at 95% confidence, test the hypotheses.
A law enforcement agent believes that at least 88% of the drivers stopped on Saturday nights
for speeding are under the influence of alcohol. A sample of 66 drivers who were stopped for
speeding on a Saturday night was taken. Eighty percent of the drivers in the sample were under
the influence of alcohol.
a. State the null and alternative hypotheses.
b. Compute the test statistic.
c. Using the p-value approach, test the hypotheses at the .05 level of significance.
Ahmadi, 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 64 of the automobiles. The sample
showed an average of 51.5 miles per gallon with a standard deviation of 4 miles per gallon.
a. Formulate the hypotheses to determine whether or not the manufacturer's advertising
campaign is legitimate.
b. Compute the test statistic.
c. What is the p-value associated with the sample results and what is your conclusion?
Let α = .05.
How many tissues should the Kimberly Clark Corporation package of Kleenex contain?
Researchers determined that 60 tissues is the mean number of tissues used during a cold.
Suppose a random sample of 100 Kleenex users yielded the following data on the number of
tissues used during a cold: = 52, S = 22. Give the null and alternative hypotheses to determine
if the number of tissues used during a cold is less than 60.
Last year, 50% of MNM, Inc. employees were female. It is believed that there has been a
reduction in the percentage of females in the company. This year, in a random sample of 400
employees, 180 were female.
a. Give the null and the alternative hypotheses.
b. At 95% confidence using the critical value approach, determine if there has been a
significant reduction in the proportion of females.
c. Show that the p-value approach results in the same conclusion as that of Part b.
Quiz
Chapter 9
When individuals in a sample of 150 were asked whether or not they supported capital
punishment, the following information was obtained.
Do you support capital punishment?
Number of individuals
Yes
40
No
60
No Opinion
50
We are interested in determining whether or not the opinions of the individuals (as to
Yes, No, and No Opinion) are uniformly distributed.
The expected frequency for each group is 50
The calculated value for the test statistic equals 4
The number of degrees of freedom associated with this problem is 2
The p-value is
Is the distribution uniform? distribution is uniform
Last school year, in the school of Business Administration, 30% were Accounting majors,
24% Management majors, 26% Marketing majors, and 20% Economics majors. A
sample of 300 students taken from this year's students of the school showed the following
number of students in each major:
We want to see if there has been a significant change in the number of students in each
major.
Accounting
83
Management
68
Marketing
85
Economics
64
Total
300
Compute the test statistic.
Has there been any significant change in the number of students in each major between
the last school year and this school year. Use the p-value approach and let α = .05.
There has been no significant change in the number of students in each
major between the last school year and this school year.
The p-value is 0.645.
The table below gives beverage preferences for random samples of teens and adults.
We are asked to test for independence between age (i.e., adult and teen) and drink
preferences.
With a .05 level of significance, the critical value for the test is_ 7.815__.
The expected number of adults who prefer coffee is_ 150__
The test statistic for this test of independence is__ 62.5_
Is drink preference independence of age?
Chapter 8
A sample of 10 items provides a sample standard deviation of 6. Compute 95% interval
estimate of the population variance.
A company claims that the standard deviation in their delivery time is less than 5 days. A
sample of 27 past customers is taken. The average delivery time in the sample was 14 days
with a standard deviation of 4.5 days. At 95% confidence, test the company's claim. [hint:
critical value=15.379]
A sample of 22 bottles of soft drink showed a variance of 0.64 in their contents. At 95%
confidence, determine whether or not the standard deviation of the population is
significantly different from 0.7. Use the critical value approach. [hint: critical value:
10.283, 35.479]
From a sample of 10 houses, the average house price in Kuching RM360,000 with a
sample standard deviation of RM6,000. Based on a sample of 9 houses, the average house
price in Samarahan is RM360,000 with a sample standard deviation of RM10,000. Are
the two population variance equal, at 10% significance level?