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Inference about a Population
Mean
BPS chapter 18
© 2006 W.H. Freeman and Company
Distributions
When we replace  with s in our standardized test statistic formula, the
distribution of the test statistic
a)
b)
c)
Changes to a t-distribution.
Stays as the standardized Normal.
Changes to an estimate of the Normal distribution.
Distributions (answer)
When we replace  with s in our standardized test statistic formula, the
distribution of the test statistic
a)
b)
c)
Changes to a t-distribution.
Stays as the standardized Normal.
Changes to an estimate of the Normal distribution.
t-distribution
When do we use the t-distribution to make inference about ?
a)
b)
c)
d)
When we know  but  is unknown.
When we have a very large sample size.
When the data are very skewed or when outliers are present.
When we do not know  or .
t-distribution (answer)
When do we use the t-distribution to make inference about ?
a)
b)
c)
d)
When we know  but  is unknown.
When we have a very large sample size.
When the data are very skewed or when outliers are present.
When we do not know  or .
Standard error
s
What is NOT true about the standard error of the sample mean, n ?

a)
b)
c)
d)
It estimates n .
We need  when computing it.
We use it when we do not know .
It is an estimate for the standard deviation of
x.
Standard error (answer)
s
What is NOT true about the standard error of the sample mean, n ?

a)
b)
c)
d)
It estimates n .
We need  when computing it.
We use it when we do not know .
It is an estimate for the standard deviation of
x.
t-distribution
Which of the following graphs could be a t-distribution?
a)
b)
c)
Plot A
Plot B
Plot C
t-distribution (answer)
Which of the following graphs could be a t-distribution?
a)
b)
c)
Plot A
Plot B
Plot C
t-distribution
The following graph shows three different density curves. If Line B
shows a t-distribution, which line could NOT show a normal
distribution?
a)
b)
Line A
Line C
t-distribution (answer)
The following graph shows three different density curves. If Line B
shows a t-distribution, which line could NOT show a normal
distribution?
a)
b)
Line A
Line C
t-table
Many people believe that playing chess can have a positive effect on
reading ability. A sample of n = 8 chess players was taken. After
participating in a comprehensive chess program, each individual
was given a post-test to gauge reading ability. If we want to create
a 95% confidence interval for the true mean score, what is the
correct t* for this problem?
a)
b)
c)
1.645
2.306
2.365
t-table (answer)
Many people believe that playing chess can have a positive effect on
reading ability. A sample of n = 8 chess players was taken. After
participating in a comprehensive chess program, each individual
was given a post-test to gauge reading ability. If we want to create
a 95% confidence interval for the true mean score, what is the
correct t* for this problem?
a)
b)
c)
1.645
2.306
2.365
t-table
Many people believe that playing chess can have a positive effect on
reading ability. A sample of n = 53 chess players was taken. After
participating in a comprehensive chess program, each individual
was given a post-test to gauge reading ability. If we want to create
a 90% confidence interval for the true mean score, what is the
correct t* for this problem?
a)
b)
c)
d)
1.645
1.671
1.676
2.009
t-table (answer)
Many people believe that playing chess can have a positive effect on
reading ability. A sample of n = 53 chess players was taken. After
participating in a comprehensive chess program, each individual
was given a post-test to gauge reading ability. If we want to create
a 90% confidence interval for the true mean score, what is the
correct t* for this problem?
a)
b)
c)
d)
1.645
1.671
1.676
2.009
Hypothesis testing
A tire manufacturer claims that one particular type of tire will last at
least 50,000 miles. A group of angry customers does not believe
this is so. They take a sample of 14 tires and want to test if the
mean mileage of the tires is really less than 50,000. What set of
hypotheses are they interested in testing?
a)
b)
c)
d)
Hypothesis testing (answer)
A tire manufacturer claims that one particular type of tire will last at
least 50,000 miles. A group of angry customers does not believe
this is so. They take a sample of 14 tires and want to test if the
mean mileage of the tires is really less than 50,000. What set of
hypotheses are they interested in testing?
a)
b)
c)
d)
Hypothesis testing
A tire manufacturer claims that one particular type of tire will last at
least 50,000 miles. A group of angry customers does not believe
this is so. They take a sample of 14 tires and want to test if the
mean mileage of the tires is really less than 50,000. If the test
statistic came out to be –3.006, what is the range for the
corresponding P-value?
a) 0.001 < P-value < 0.0025
b)
0.0025 < P-value < 0.005
c)
0.005 < P-value < 0.01
d) Impossible to determine since no negative numbers on the table.
Hypothesis testing (answer)
A tire manufacturer claims that one particular type of tire will last at
least 50,000 miles. A group of angry customers does not believe
this is so. They take a sample of 14 tires and want to test if the
mean mileage of the tires is really less than 50,000. If the test
statistic came out to be –3.006, what is the range for the
corresponding P-value?
a) 0.001 < P-value < 0.0025
b)
0.0025 < P-value < 0.005
c)
0.005 < P-value < 0.01
d) Impossible to determine since no negative numbers on the table.
Hypothesis testing
A tire manufacturer claims that one particular type of tire will last at
least 50,000 miles. A group of angry customers does not believe
this is so. They take a sample of 14 tires and want to test if the
mean mileage of the tires is really less than 50,000. If 0.0100 < Pvalue < 0.0200, what decision should be made if testing at the  =
0.05 level?
a)
b)
c)
d)
Reject H0 and conclude that the tires were not performing as
claimed.
Reject H0 and conclude that the mean tire life really is 50,000 miles.
Do not reject H0 and conclude that the tires were not performing as
claimed.
Do not reject H0 and conclude that the mean tire life really is 50,000
miles.
Hypothesis testing (answer)
A tire manufacturer claims that one particular type of tire will last at
least 50,000 miles. A group of angry customers does not believe
this is so. They take a sample of 14 tires and want to test if the
mean mileage of the tires is really less than 50,000. If 0.0100 < Pvalue < 0.0200, what decision should be made if testing at the  =
0.05 level?
a)
b)
c)
d)
Reject H0 and conclude that the tires were not performing as
claimed.
Reject H0 and conclude that the mean tire life really is 50,000 miles.
Do not reject H0 and conclude that the tires were not performing as
claimed.
Do not reject H0 and conclude that the mean tire life really is 50,000
miles.
t procedures
A student wanted to assess the average time spent studying for his
most recent exam taken in class. He asked the first 45 students
who came to class how much time they spent and recorded the
values. He then used this information to calculate a 95%
confidence interval for the mean time spent by all students. Was
this an appropriate use of the t procedure for a confidence interval?.
a)
b)
c)
d)
Yes, because we can assume that the time spent studying was
normally distributed.
Yes, because the 45 students is a big enough sample to invoke the
CLT.
No, because time spent studying is not a quantitative variable.
No, because there was no randomization in choosing the sample.
t procedures (answer)
A student wanted to assess the average time spent studying for his
most recent exam taken in class. He asked the first 45 students
who came to class how much time they spent and recorded the
values. He then used this information to calculate a 95%
confidence interval for the mean time spent by all students. Was
this an appropriate use of the t procedure for a confidence interval?.
a)
b)
c)
d)
Yes, because we can assume that the time spent studying was
normally distributed.
Yes, because the 45 students is a big enough sample to invoke the
CLT.
No, because time spent studying is not a quantitative variable.
No, because there was no randomization in choosing the
sample.
t procedures
The same student from the previous question also wanted to test if the
mean score for all students taking the exam was equal to 80%. He
took a simple random sample of size n = 18 students from his class.
After taking his sample, he created the following histogram:
Should he continue his analysis and perform this test of hypothesis?
a)
b)
No, because the data are not bell-shaped and the sample size is not
large enough to use the CLT.
Yes, because he used a random method with which to collect his
data.
t procedures (answer)
The same student from the previous question also wanted to test if the
mean score for all students taking the exam was equal to 80%. He
took a simple random sample of size n = 18 students from his class.
After taking his sample, he created the following histogram:
Should he continue his analysis and perform this test of hypothesis?
a)
b)
No, because the data are not bell-shaped and the sample size
is not large enough to use the CLT.
Yes, because he used a random method with which to collect his
data.
t procedures
The following histogram represents the yearly advertising budgets (in
millions of dollars) of 21 randomly selected companies. A statistics
student wants to create a confidence interval for the mean
advertising budget of all companies. By looking at the histogram, is
the use of a t procedure appropriate in this case?
a)
b)
c)
d)
Yes, because data were from an experiment.
Yes, because the sample size is large enough.
No, because the data are skewed and have outliers.
No, because we did not repeat the sampling enough times.
t procedures (answer)
The following histogram represents the yearly advertising budgets (in
millions of dollars) of 21 randomly selected companies. A statistics
student wants to create a confidence interval for the mean
advertising budget of all companies. By looking at the histogram, is
the use of a t procedure appropriate in this case?
a)
b)
c)
d)
Yes, because data were from an experiment.
Yes, because the sample size is large enough.
No, because the data are skewed and have outliers.
No, because we did not repeat the sampling enough times.
t procedures
A train operator claims that her trains are never more than 7 minutes
late. A commuter takes a simple random sample of trains arriving at
her local station of size n = 21 and records their estimated arrival
time and the actual arrival time. She creates the following stemplot
for these data:
Based on the stemplot, should she continue with her use of the tprocedure?
a) No, because the sample size is not large enough.
b) Yes, because the data show no strong skewness or outliers.
t procedures (answer)
A train operator claims that her trains are never more than 7 minutes
late. A commuter takes a simple random sample of trains arriving at
her local station of size n = 21 and records their estimated arrival
time and the actual arrival time. She creates the following stemplot
for these data:
Based on the stemplot, should she continue with her use of the tprocedure?
a) No, because the sample size is not large enough.
b) Yes, because the data show no strong skewness or outliers.