Statistics Chapter 8 Notes
Content
Name:_______________________________________
Statistics Chapter 8 Notes
Hypothesis testing
Sec. 8.1 - Introduction
What is hypothesis testing?
In hypothesis testing, the researcher must define
________________________, state the particular
___________________________________________, give the _________
__________, select a _________ from the population, collect the ________,
perform the _____________________ required for the statistical test, and
______________________ ___________.
The three methods used to test hypotheses are;
1)
2)
3)
Sec. 8.2 - Steps in Hypothesis Testing – Traditional Method
A statistical hypothesis is …
The null hypothesis is …
The null hypothesis can be symbolized by ____.
The alternative hypothesis is …
The alternative hypothesis can be symbolized by ____.
Name:_______________________________________
Ex. 1
State the null and alternative hypothesis for each conjecture.
a) A researcher thinks that is expectant mothers use vitamin pills ,
the birth weight of the babies will increase. The average birth
weight of the population is 8.6 pounds.
H0:
H1:
b) An engineer hypothesizes that the mean number of defects can
be decreased in a manufacturing process of compact disks by
using robots instead of humans for certain tasks. The mean
number of defective disks per 1000 is 18.
H0:
H1:
c) A psychologist feels that playing soft music during a test will
change the results of the test. The psychologist is not sure
whether the grades will be higher of lower. In the past, the mean
of the scores was 73.
H0:
H1:
What does a statistical test do?
The numerical value obtained from a statistical test is called the
____________________.
Name:_______________________________________
Type 1 error occurs …
This would be comparable to what result in a criminal trial?
Type II error occurs …
This would be comparable to what result in a criminal trial?
The level of significance is …
This probability is symbolized by the greek letter ______.
The critical value separates the __________________ from the
_________________.
The critical region (rejection region) is …
The noncritical region (nonrejection region) is…
A one-tailed test indicates that the null hypothesis should
be_____________ when the test value is in the critical region on one side
of the mean. A one-tailed test is either a ___________________ or a
______________________, depending on the direction of the inequality of
the alternative hypothesis.
Name:_______________________________________
In a two-tailed test, the null hypothesis should be _____________ when
the test value is in either of the two critical regions.
Procedure for finding the Critical Values for Specific
Values, using
table E
Step 1
Draw the figures and indicate the appropriate area.
a)
If the test is __________________________, the critical
region, with an area equal to _______, will be on the
___________ side of the mean.
b)
If the test is __________________________, the critical
region, with an area equal to _______, will be on the
___________ side of the mean.
c)
If the test is __________________________, ___ must be
divided by 2; one-half of the area will be to the
____________ of the mean, and one-half will be to
the ___________ of the mean.
Step 2
For a one-tailed test, subtract the area (equivalent to _________) in the
critical region from ________. For a two-tailed test, subtract the area
(equivalent to _________) from __________.
Step 3
Find the area in table E corresponding to the value obtained in step 2.
If the exact value cannot be found in the table, __________________.
Step 4
Find the _____________ that corresponds to the area. This will be the
_____________ _______________.
Step 5
Determine the sign of the critical value for a one-tailed test.
a) If the test is left-tailed, the critical value will be
_______________.
b) If the test is right-tailed, the critical value will be
________________.
For a two-tailed test, one value will be _______________ and the
other _______________.
Ex. 2.
Using Table E, find the critical value for each situation and draw the
appropriate figure, showing the critical region.
a) A left-tailed test with = 0.10.
Name:_______________________________________
b) A two-tailed test with
= 0.02.
c) A right-tailed test with
= 0.005.
Procedure for the Traditional Method of Solving Hypothesis-Testing
Problems
Step 1
Step 2
Step 3
Step 4
Step 5
Sec. 8.3 - Z test for Mean
The z test is…
It can be used when …
Name:_______________________________________
The formula for the z test is z =
Ex. 3
A researcher reports that the average salary of assistant professors is
more than $42,000. A sample if 30 assistant professors has a mean
salary of $43,260. At = 0.05, test the claim that assistant professors
earn more than $42,000 a year. The standard deviation of the
population is $5230.
Ex. 4
A researcher claims that the average cost of men’s athletic shoes is
less than $80. He selects a random sample of 36 pairs of shoes from a
catalog and finds the following costs (in dollars). Is there enough
evidence to support the researcher’s claim at = 0.10?
60
120
75
Ex. 5
70
90
60
75
75
90
55
85
90
80
80
60
55
60
95
50
110
110
40
65
85
80
80
45
70
85
90
50
85
70
95
45
70
Name:_______________________________________
The Medical Rehab. Ed. Foundation reports that the average cost of
rehab for a stroke victim is $24,672. To see if the average cost of
rehab is different at a particular hospital, a researcher selects a
random sample of 35 stroke victims at the hospital and finds that the
average cost of their rehab is $25,226. The standard deviation of the
population is $3251. At = 0.01, can it be concluded that the average
cost of stroke rehab at a particular hospital is different from $24,672?
What is a P-value?
Procedure for P-Value method of solving hypothesis-testing problems
Step 1
Step 2
Step 3
Step 4
Step 5
Ex. 6
A researcher wishes to test the claim that average age of lifeguards in
Ocean City is greater than 24 years. She selects a sample of 36 guards
and finds the mean of the sample to be 24.7 years, with a standard
deviation of 2 years. Is there enough evidence to support the claim at
= 0.05? Use the P-value method.
Name:_______________________________________
Ex. 7
A researcher claims that the average wind speed in a certain city is 8
miles per hour. A sample of 32 days has an average wind speed of 8.2
miles per hour. The standard deviation of the sample is 0.6 miles per
hour. At = 0.05, is there enough evidence to reject the claim? Use
the P-value method.
Decision Rule When using a P-value:
If P-value
, ________________________________________
If P-value > , ________________________________________
Guidelines for P-Values:
If P-value _________________, reject the null hypothesis.
If P-value ________________ but __________________, reject the null
hypothesis.
If P-value _______ but_____________ , consider the consequences of type
1 error before rejecting the null hypothesis.
If P-value _____________, do not reject the null hypothesis.
Sec. 8.4 - t test for Mean
When do you need to use the t test?
The formula for the t test is t =
Ex. 8
Name:_______________________________________
Find the critical t value for
= 0.05 with d.f. = 16 for a right-tailed test.
Ex. 9
Find the critical t value for
= 0.01 with d.f. = 22 for a left-tailed test.
Ex. 10
Find the critical t value for
= 0.10 with d.f. = 18 for a two-tailed test.
Ex. 11
Find the critical t value for
= 0.05 with d.f. = 28 for a right-tailed test.
Ex. 12
A job placement director claims that the average starting salary for
nurses is $24,000. A sample of 10 nurses’ salaries has a mean of
$23,450 and a standard deviation of $400. Is there enough evidence
to reject the director’s claim at = 0.05?
Ex. 13
An educator claims that the average salary of substitute teachers in
school districts in Allegheny County, Pennsylvania, is less then $60 a
day. A random sample of 8 school districts is selected, and the daily
salaries (in dollars) are shown. Is there enough evidence to support
the educator’s claim at = 0.10?
60
56
60
55
70
55
60
55
Name:_______________________________________
Ex. 14
Find the P-value when the t test value is 2.056, the sample size is 11,
and the test is right-tailed.
Ex. 15
Find the P-value when the t test value is 2.983, the sample size is 6,
and the test is two-tailed.
Ex. 16
A physician claims that joggers’ maximal volume oxygen uptake is
greater than the average of all adults. A sample of 15 joggers has a
mean of 40.6 milliliters per kilogram and a standard deviation of 6
ml/kg. If the average of all adults is 36.7 ml/kg is there enough
evidence to support the physician’s claim at = 0.05?
Sec. 8.5 – z test for Proportion
Write out the formula for the z test for Proportions.
Ex. 17
Name:_______________________________________
An educator estimates that the dropout rate for seniors at high schools
in Ohio is 15%. Last year, 38 seniors from a random sample of 200
Ohio seniors withdrew. At = 0.05, is there enough evidence to reject
the educator’s claim?
Ex. 18
A telephone company representative estimates that 40% of its
customers have call-waiting service. To test this hypothesis, she
selected a sample of 100 customers and found that 37% had call
waiting. At = 0.01, is there enough evidence to reject the claim?
Ex. 19
A statistician read that at least 77% of the population oppose replacing
$1 bills with $1 coins. To see if the claim is valid, the statistician
selected a sample of 80 people and found that 55 were opposed to
replacing the $1 bills. At = 0.01, test the claim that at least 77% of
the population are opposed to the change.
Ex. 20
An attorney claims that more than 25% of all lawyers advertise. A
sample of 200 lawyers in a certain city showed that 63 had used some
Name:_______________________________________
form of advertising. At
= 0.05, is there enough evidence to support
the attorney’s claim? Use the P-value method.
Sec. 8.6 -
Test for a Variance or Standard Deviation
Ex. 21
Find the critical chi-square value for 15 degrees of freedom when
=
0.05 and the test is right-tailed.
Ex. 22
Find the critical chi-square value for 10 degrees of freedom when
=
0.05 and the test is left-tailed.
Ex. 23
Find the critical chi-square value for 22 degrees of freedom when
0.05 and the test is two-tailed.
Write out the formula for the Chi-square Test for a Single variance.
Assumptions for the Chi-Square Test for a Single Variance
1)
2)
3)
=
Name:_______________________________________
Ex. 24
An instructor wishes to see whether the variation in scores of the 23
students in her class is less than the variance of the population. The
variance of the class is 198. Is there enough evidence to support the
claim that the variation of the students is less than the population
variance ( =225) at = 0.05? Assume that the scores are normally
distributed.
Ex. 25
A hospital administrator believes that the standard deviation of the
number of people using outpatient surgery per day is greater than 8. A
random sample of 15 days is selected. The data are shown. At =
0.10, is there enough evidence to support the administrator’s claim?
Assume the variable is normally distributed.
25
30
5
15
18
42
16
9
10
12
12
38
8
14
27
Ex. 26
A cigarette manufacturer wishes to test the claim that the variance of
the nicotine content of its cigarettes is 0.644. Nicotine content is
measured in milligrams, and assume that it is normally distributed. A
sample of 20 cigarettes has a standard deviation of 1.00 milligram. At
= 0.05, is there enough evidence to reject the manufacturer’ s claim?
Name:_______________________________________
Ex. 27
Find the P-value when
= 19.274, n=8, and the test is right-tailed.
Ex. 28
Find the P-value when
= 3.823, n=13, and the test is left-tailed.
*When the
test is two-tailed, both interval values must be
___________________.
Ex. 29
A researcher knows from past studies that the standard deviation of
the time it takes to inspect a car is 16.8 minutes. A sample of 24 cars
is selected and inspected. The standard deviation was 12.5 minutes.
At = 0.05, can it be concluded that the standard deviation has
changed? Use the P-value method.
Sec. 8.7 – Additional Topics Regarding Hypothesis Testing
When the null hypothesis is rejected the confidence interval for the
mean _____________
contain the hypothesized mean.
When the null hypothesis is not rejected the confidence interval for the
mean _________
contain the hypothesized mean.
Ex. 30
Sugar is packed in 5-pound bags. An inspector suspects the bags may
not contain 5 pounds. A sample of 50 bags produces a mean of 4.6
pounds and a standard deviation of 0.7 pounds. Is there enough
evidence to conclude that the bags do not contain 5 pounds as stated
at = 0.05? Also, find the 95% confidence interval of the true mean.
Name:_______________________________________
Ex. 31
A researcher claims that adult hogs fed a special diet will have an
average weight of 200 pounds. A sample of 10 hogs has an average
weight of 198.2 pounds and a standard deviation of 3.3 pounds. At =
0.05, can the claim be rejected? Also, find the 95% confidence interval
of the true mean.
Sec. 8.8 - Summary
Please summarize the chapter.
Statistics Chapter 8 Notes
Hypothesis testing
Sec. 8.1 - Introduction
What is hypothesis testing?
In hypothesis testing, the researcher must define
________________________, state the particular
___________________________________________, give the _________
__________, select a _________ from the population, collect the ________,
perform the _____________________ required for the statistical test, and
______________________ ___________.
The three methods used to test hypotheses are;
1)
2)
3)
Sec. 8.2 - Steps in Hypothesis Testing – Traditional Method
A statistical hypothesis is …
The null hypothesis is …
The null hypothesis can be symbolized by ____.
The alternative hypothesis is …
The alternative hypothesis can be symbolized by ____.
Name:_______________________________________
Ex. 1
State the null and alternative hypothesis for each conjecture.
a) A researcher thinks that is expectant mothers use vitamin pills ,
the birth weight of the babies will increase. The average birth
weight of the population is 8.6 pounds.
H0:
H1:
b) An engineer hypothesizes that the mean number of defects can
be decreased in a manufacturing process of compact disks by
using robots instead of humans for certain tasks. The mean
number of defective disks per 1000 is 18.
H0:
H1:
c) A psychologist feels that playing soft music during a test will
change the results of the test. The psychologist is not sure
whether the grades will be higher of lower. In the past, the mean
of the scores was 73.
H0:
H1:
What does a statistical test do?
The numerical value obtained from a statistical test is called the
____________________.
Name:_______________________________________
Type 1 error occurs …
This would be comparable to what result in a criminal trial?
Type II error occurs …
This would be comparable to what result in a criminal trial?
The level of significance is …
This probability is symbolized by the greek letter ______.
The critical value separates the __________________ from the
_________________.
The critical region (rejection region) is …
The noncritical region (nonrejection region) is…
A one-tailed test indicates that the null hypothesis should
be_____________ when the test value is in the critical region on one side
of the mean. A one-tailed test is either a ___________________ or a
______________________, depending on the direction of the inequality of
the alternative hypothesis.
Name:_______________________________________
In a two-tailed test, the null hypothesis should be _____________ when
the test value is in either of the two critical regions.
Procedure for finding the Critical Values for Specific
Values, using
table E
Step 1
Draw the figures and indicate the appropriate area.
a)
If the test is __________________________, the critical
region, with an area equal to _______, will be on the
___________ side of the mean.
b)
If the test is __________________________, the critical
region, with an area equal to _______, will be on the
___________ side of the mean.
c)
If the test is __________________________, ___ must be
divided by 2; one-half of the area will be to the
____________ of the mean, and one-half will be to
the ___________ of the mean.
Step 2
For a one-tailed test, subtract the area (equivalent to _________) in the
critical region from ________. For a two-tailed test, subtract the area
(equivalent to _________) from __________.
Step 3
Find the area in table E corresponding to the value obtained in step 2.
If the exact value cannot be found in the table, __________________.
Step 4
Find the _____________ that corresponds to the area. This will be the
_____________ _______________.
Step 5
Determine the sign of the critical value for a one-tailed test.
a) If the test is left-tailed, the critical value will be
_______________.
b) If the test is right-tailed, the critical value will be
________________.
For a two-tailed test, one value will be _______________ and the
other _______________.
Ex. 2.
Using Table E, find the critical value for each situation and draw the
appropriate figure, showing the critical region.
a) A left-tailed test with = 0.10.
Name:_______________________________________
b) A two-tailed test with
= 0.02.
c) A right-tailed test with
= 0.005.
Procedure for the Traditional Method of Solving Hypothesis-Testing
Problems
Step 1
Step 2
Step 3
Step 4
Step 5
Sec. 8.3 - Z test for Mean
The z test is…
It can be used when …
Name:_______________________________________
The formula for the z test is z =
Ex. 3
A researcher reports that the average salary of assistant professors is
more than $42,000. A sample if 30 assistant professors has a mean
salary of $43,260. At = 0.05, test the claim that assistant professors
earn more than $42,000 a year. The standard deviation of the
population is $5230.
Ex. 4
A researcher claims that the average cost of men’s athletic shoes is
less than $80. He selects a random sample of 36 pairs of shoes from a
catalog and finds the following costs (in dollars). Is there enough
evidence to support the researcher’s claim at = 0.10?
60
120
75
Ex. 5
70
90
60
75
75
90
55
85
90
80
80
60
55
60
95
50
110
110
40
65
85
80
80
45
70
85
90
50
85
70
95
45
70
Name:_______________________________________
The Medical Rehab. Ed. Foundation reports that the average cost of
rehab for a stroke victim is $24,672. To see if the average cost of
rehab is different at a particular hospital, a researcher selects a
random sample of 35 stroke victims at the hospital and finds that the
average cost of their rehab is $25,226. The standard deviation of the
population is $3251. At = 0.01, can it be concluded that the average
cost of stroke rehab at a particular hospital is different from $24,672?
What is a P-value?
Procedure for P-Value method of solving hypothesis-testing problems
Step 1
Step 2
Step 3
Step 4
Step 5
Ex. 6
A researcher wishes to test the claim that average age of lifeguards in
Ocean City is greater than 24 years. She selects a sample of 36 guards
and finds the mean of the sample to be 24.7 years, with a standard
deviation of 2 years. Is there enough evidence to support the claim at
= 0.05? Use the P-value method.
Name:_______________________________________
Ex. 7
A researcher claims that the average wind speed in a certain city is 8
miles per hour. A sample of 32 days has an average wind speed of 8.2
miles per hour. The standard deviation of the sample is 0.6 miles per
hour. At = 0.05, is there enough evidence to reject the claim? Use
the P-value method.
Decision Rule When using a P-value:
If P-value
, ________________________________________
If P-value > , ________________________________________
Guidelines for P-Values:
If P-value _________________, reject the null hypothesis.
If P-value ________________ but __________________, reject the null
hypothesis.
If P-value _______ but_____________ , consider the consequences of type
1 error before rejecting the null hypothesis.
If P-value _____________, do not reject the null hypothesis.
Sec. 8.4 - t test for Mean
When do you need to use the t test?
The formula for the t test is t =
Ex. 8
Name:_______________________________________
Find the critical t value for
= 0.05 with d.f. = 16 for a right-tailed test.
Ex. 9
Find the critical t value for
= 0.01 with d.f. = 22 for a left-tailed test.
Ex. 10
Find the critical t value for
= 0.10 with d.f. = 18 for a two-tailed test.
Ex. 11
Find the critical t value for
= 0.05 with d.f. = 28 for a right-tailed test.
Ex. 12
A job placement director claims that the average starting salary for
nurses is $24,000. A sample of 10 nurses’ salaries has a mean of
$23,450 and a standard deviation of $400. Is there enough evidence
to reject the director’s claim at = 0.05?
Ex. 13
An educator claims that the average salary of substitute teachers in
school districts in Allegheny County, Pennsylvania, is less then $60 a
day. A random sample of 8 school districts is selected, and the daily
salaries (in dollars) are shown. Is there enough evidence to support
the educator’s claim at = 0.10?
60
56
60
55
70
55
60
55
Name:_______________________________________
Ex. 14
Find the P-value when the t test value is 2.056, the sample size is 11,
and the test is right-tailed.
Ex. 15
Find the P-value when the t test value is 2.983, the sample size is 6,
and the test is two-tailed.
Ex. 16
A physician claims that joggers’ maximal volume oxygen uptake is
greater than the average of all adults. A sample of 15 joggers has a
mean of 40.6 milliliters per kilogram and a standard deviation of 6
ml/kg. If the average of all adults is 36.7 ml/kg is there enough
evidence to support the physician’s claim at = 0.05?
Sec. 8.5 – z test for Proportion
Write out the formula for the z test for Proportions.
Ex. 17
Name:_______________________________________
An educator estimates that the dropout rate for seniors at high schools
in Ohio is 15%. Last year, 38 seniors from a random sample of 200
Ohio seniors withdrew. At = 0.05, is there enough evidence to reject
the educator’s claim?
Ex. 18
A telephone company representative estimates that 40% of its
customers have call-waiting service. To test this hypothesis, she
selected a sample of 100 customers and found that 37% had call
waiting. At = 0.01, is there enough evidence to reject the claim?
Ex. 19
A statistician read that at least 77% of the population oppose replacing
$1 bills with $1 coins. To see if the claim is valid, the statistician
selected a sample of 80 people and found that 55 were opposed to
replacing the $1 bills. At = 0.01, test the claim that at least 77% of
the population are opposed to the change.
Ex. 20
An attorney claims that more than 25% of all lawyers advertise. A
sample of 200 lawyers in a certain city showed that 63 had used some
Name:_______________________________________
form of advertising. At
= 0.05, is there enough evidence to support
the attorney’s claim? Use the P-value method.
Sec. 8.6 -
Test for a Variance or Standard Deviation
Ex. 21
Find the critical chi-square value for 15 degrees of freedom when
=
0.05 and the test is right-tailed.
Ex. 22
Find the critical chi-square value for 10 degrees of freedom when
=
0.05 and the test is left-tailed.
Ex. 23
Find the critical chi-square value for 22 degrees of freedom when
0.05 and the test is two-tailed.
Write out the formula for the Chi-square Test for a Single variance.
Assumptions for the Chi-Square Test for a Single Variance
1)
2)
3)
=
Name:_______________________________________
Ex. 24
An instructor wishes to see whether the variation in scores of the 23
students in her class is less than the variance of the population. The
variance of the class is 198. Is there enough evidence to support the
claim that the variation of the students is less than the population
variance ( =225) at = 0.05? Assume that the scores are normally
distributed.
Ex. 25
A hospital administrator believes that the standard deviation of the
number of people using outpatient surgery per day is greater than 8. A
random sample of 15 days is selected. The data are shown. At =
0.10, is there enough evidence to support the administrator’s claim?
Assume the variable is normally distributed.
25
30
5
15
18
42
16
9
10
12
12
38
8
14
27
Ex. 26
A cigarette manufacturer wishes to test the claim that the variance of
the nicotine content of its cigarettes is 0.644. Nicotine content is
measured in milligrams, and assume that it is normally distributed. A
sample of 20 cigarettes has a standard deviation of 1.00 milligram. At
= 0.05, is there enough evidence to reject the manufacturer’ s claim?
Name:_______________________________________
Ex. 27
Find the P-value when
= 19.274, n=8, and the test is right-tailed.
Ex. 28
Find the P-value when
= 3.823, n=13, and the test is left-tailed.
*When the
test is two-tailed, both interval values must be
___________________.
Ex. 29
A researcher knows from past studies that the standard deviation of
the time it takes to inspect a car is 16.8 minutes. A sample of 24 cars
is selected and inspected. The standard deviation was 12.5 minutes.
At = 0.05, can it be concluded that the standard deviation has
changed? Use the P-value method.
Sec. 8.7 – Additional Topics Regarding Hypothesis Testing
When the null hypothesis is rejected the confidence interval for the
mean _____________
contain the hypothesized mean.
When the null hypothesis is not rejected the confidence interval for the
mean _________
contain the hypothesized mean.
Ex. 30
Sugar is packed in 5-pound bags. An inspector suspects the bags may
not contain 5 pounds. A sample of 50 bags produces a mean of 4.6
pounds and a standard deviation of 0.7 pounds. Is there enough
evidence to conclude that the bags do not contain 5 pounds as stated
at = 0.05? Also, find the 95% confidence interval of the true mean.
Name:_______________________________________
Ex. 31
A researcher claims that adult hogs fed a special diet will have an
average weight of 200 pounds. A sample of 10 hogs has an average
weight of 198.2 pounds and a standard deviation of 3.3 pounds. At =
0.05, can the claim be rejected? Also, find the 95% confidence interval
of the true mean.
Sec. 8.8 - Summary
Please summarize the chapter.
