WGU Academy Statistics Module 12 With
Diagrams for easier understanding
The U.S. government sponsored the National Health and Nutrition Examination Survey
(NHANES). For the survey, a random sample of 712 males between 20 and 29 years of
age and a random sample of 1,001 males over the age of 75 were chosen. The weight
of each of the males was recorded (in kg).
A
Do the data provide evidence that the younger male population weighs more (on
average) than the older male population? (Comment: Note that here the data are given
VI
in a summarized form, unlike the previous problem, where the raw data were given.)
TU
IS
Note that we defined the younger age group and the older age group as population 1
OM
and population 2, respectively. We defined μ 1 and μ 2 as the mean weight of
population 1 and population 2, respectively.
Step 1:
Since we want to test whether the older age group (population 2) weighs less on
NA
average than the younger age group (population 1), we are testing:
H0:μ1−μ2=0
Ha:μ1−μ2>0
or equivalently,
JP
H0:μ1=μ2
Ha:μ1>μ2
Step 2:
We can safely use the two-sample t-test in this case since:
The samples are independent, since each of the samples was chosen at random.
Both sample sizes are very large (712 and 1,001). Therefore we can proceed
regardless of whether the pop
,Which of the following is the correct interpretation of the fact that the P-value is 0?
It is nearly impossible that younger males weigh, on average, more than older males.
It would be nearly impossible to get results like observed in this study or more extreme
(i.e., a difference between averages of 4.9 or larger) assuming no difference in mean
weight between younger and older males.
It would be nearly impossible to get results like observed in this study or more extreme
A
(i.e., a difference between averages of 4.9 or larger).
VI
It is nearly impossible that there is no difference between in mean weight between
younger and older males.
TU
It would be nearly impossible to get results like observed in this study or more extreme
(i.e., a difference between averages of 4.9 or larger) assuming that younger males
weigh, on average, more than older males.
It would be nearly impossible to get results like observed in this study or more extreme
IS
(i.e., a difference between averages of 4.9 or larger) assuming no difference in mean
weight between younger and older males.
-The p-value is the probability of getting results like that observed (or more extreme)
OM
assuming that the null hypothesis is true.
Confidence Interval for μ1 − μ2
So far we've discussed the two-sample t-test. It checks if there is enough evidence in
NA
the data to reject the claim μ1−μ2=0 (or equivalently, that μ1=μ2) in favor of one of the
three alternatives.
If we would like to estimate μ1−μ2, we can use the natural point estimate, y1¯− y2¯, or
preferably, a 95% confidence interval. This will provide us with a set of plausible values
for the difference between the population means μ1−μ2.
JP
In particular, if the test has rejected Ho:μ1−μ2=0, a confidence interval for μ1−μ2 can be
insightful. The confidence interval quantifies the effect that the categorical explanatory
variable has on the response.
the 95% confidence interval is (-3.69590, -1.49625), and that the p-value is 0.000.
,We used the fact that the p-value is so small to conclude that Ho can be rejected. We
can also use the confidence interval to reach the same conclusion since 0 falls outside
the confidence interval. In other words, since 0 is not a plausible value for μ1−μ2 we
can reject H o, which claims that μ1−μ2=0
Below you'll find three sample outputs of the two-sided two-sample t-test:
H0:μ1−μ2=0
A
vs
VI
Ha:μ1−μ2≠0
However, only one of the outputs could be correct (the other two contain an
TU
inconsistency). Your task is to decide which of the following outputs is the correct one
(Hint: No calculations are necessary in order to answer this question. Instead pay
attention to the p-value and confidence interval).
IS
Output A: p-value: 0.28995% Confidence Interval: (-5.93090, -1.78572)
Output B: p-value: 0.00395% Confidence Interval: (-13.97384, 2.89733)
OM
Output C: p-value: 0.22395% Confidence Interval: (-9.31432, 2.20505)
Which of the following is the correct output?
NA
A
B
C
JP
Output C: p-value: 0.22395% Confidence Interval: (-9.31432, 2.20505)
- this is the correct output, since it is the only one out of the three in which both the
confidence interval and p-value lead us to the same conclusion (as it should be). Note
that 0 falls inside the 95% confidence interval for μ1 - μ2, which means that Ho cannot
be rejected. Also, the p-value is large (.223) indicating that Ho cannot be rejected.
, An advertiser is experimenting with a new color scheme and conducts a study to test its
effectiveness. In which situation could they use the two-sample t-test for comparing two
population means?
They randomly expose consumers to one website when they land on their page: the old
one with the original color scheme or the new one with the updated color scheme. Then
they measure to see how much people buy.
They track each customer's spending habits in the old platform and then they change
A
the color scheme to see if spending habits go up or down for each consumer.
VI
They show consumers both options: the original color scheme or the updated color
scheme. They let consumers decide which one they like better and then select the color
scheme that most customers prefer.
TU
They randomly expose consumers to one website when they land on their page: the old
one with the original color scheme or the new one with the updated color scheme. Then
they measure to see how much people buy.
IS
College Students and Depression: A public health official is studying differences in
depression among students at two different universities. They collect a random sample
OM
of students independently from each of the two universities and administer a well known
depression inventory. A score of 5 or above indicates some depression. A score above
15 indicates that active treatment is necessary.
Sample Statistics Pictured:
NA
JP
The official conducts a two-sample t-test to determine whether these data provide
significant evidence that students at University 1 are more depressed than students at
University 2. The test statistic is t = 2.64 with a P-value 0.005.Which of the following is
an appropriate conclusion?
Diagrams for easier understanding
The U.S. government sponsored the National Health and Nutrition Examination Survey
(NHANES). For the survey, a random sample of 712 males between 20 and 29 years of
age and a random sample of 1,001 males over the age of 75 were chosen. The weight
of each of the males was recorded (in kg).
A
Do the data provide evidence that the younger male population weighs more (on
average) than the older male population? (Comment: Note that here the data are given
VI
in a summarized form, unlike the previous problem, where the raw data were given.)
TU
IS
Note that we defined the younger age group and the older age group as population 1
OM
and population 2, respectively. We defined μ 1 and μ 2 as the mean weight of
population 1 and population 2, respectively.
Step 1:
Since we want to test whether the older age group (population 2) weighs less on
NA
average than the younger age group (population 1), we are testing:
H0:μ1−μ2=0
Ha:μ1−μ2>0
or equivalently,
JP
H0:μ1=μ2
Ha:μ1>μ2
Step 2:
We can safely use the two-sample t-test in this case since:
The samples are independent, since each of the samples was chosen at random.
Both sample sizes are very large (712 and 1,001). Therefore we can proceed
regardless of whether the pop
,Which of the following is the correct interpretation of the fact that the P-value is 0?
It is nearly impossible that younger males weigh, on average, more than older males.
It would be nearly impossible to get results like observed in this study or more extreme
(i.e., a difference between averages of 4.9 or larger) assuming no difference in mean
weight between younger and older males.
It would be nearly impossible to get results like observed in this study or more extreme
A
(i.e., a difference between averages of 4.9 or larger).
VI
It is nearly impossible that there is no difference between in mean weight between
younger and older males.
TU
It would be nearly impossible to get results like observed in this study or more extreme
(i.e., a difference between averages of 4.9 or larger) assuming that younger males
weigh, on average, more than older males.
It would be nearly impossible to get results like observed in this study or more extreme
IS
(i.e., a difference between averages of 4.9 or larger) assuming no difference in mean
weight between younger and older males.
-The p-value is the probability of getting results like that observed (or more extreme)
OM
assuming that the null hypothesis is true.
Confidence Interval for μ1 − μ2
So far we've discussed the two-sample t-test. It checks if there is enough evidence in
NA
the data to reject the claim μ1−μ2=0 (or equivalently, that μ1=μ2) in favor of one of the
three alternatives.
If we would like to estimate μ1−μ2, we can use the natural point estimate, y1¯− y2¯, or
preferably, a 95% confidence interval. This will provide us with a set of plausible values
for the difference between the population means μ1−μ2.
JP
In particular, if the test has rejected Ho:μ1−μ2=0, a confidence interval for μ1−μ2 can be
insightful. The confidence interval quantifies the effect that the categorical explanatory
variable has on the response.
the 95% confidence interval is (-3.69590, -1.49625), and that the p-value is 0.000.
,We used the fact that the p-value is so small to conclude that Ho can be rejected. We
can also use the confidence interval to reach the same conclusion since 0 falls outside
the confidence interval. In other words, since 0 is not a plausible value for μ1−μ2 we
can reject H o, which claims that μ1−μ2=0
Below you'll find three sample outputs of the two-sided two-sample t-test:
H0:μ1−μ2=0
A
vs
VI
Ha:μ1−μ2≠0
However, only one of the outputs could be correct (the other two contain an
TU
inconsistency). Your task is to decide which of the following outputs is the correct one
(Hint: No calculations are necessary in order to answer this question. Instead pay
attention to the p-value and confidence interval).
IS
Output A: p-value: 0.28995% Confidence Interval: (-5.93090, -1.78572)
Output B: p-value: 0.00395% Confidence Interval: (-13.97384, 2.89733)
OM
Output C: p-value: 0.22395% Confidence Interval: (-9.31432, 2.20505)
Which of the following is the correct output?
NA
A
B
C
JP
Output C: p-value: 0.22395% Confidence Interval: (-9.31432, 2.20505)
- this is the correct output, since it is the only one out of the three in which both the
confidence interval and p-value lead us to the same conclusion (as it should be). Note
that 0 falls inside the 95% confidence interval for μ1 - μ2, which means that Ho cannot
be rejected. Also, the p-value is large (.223) indicating that Ho cannot be rejected.
, An advertiser is experimenting with a new color scheme and conducts a study to test its
effectiveness. In which situation could they use the two-sample t-test for comparing two
population means?
They randomly expose consumers to one website when they land on their page: the old
one with the original color scheme or the new one with the updated color scheme. Then
they measure to see how much people buy.
They track each customer's spending habits in the old platform and then they change
A
the color scheme to see if spending habits go up or down for each consumer.
VI
They show consumers both options: the original color scheme or the updated color
scheme. They let consumers decide which one they like better and then select the color
scheme that most customers prefer.
TU
They randomly expose consumers to one website when they land on their page: the old
one with the original color scheme or the new one with the updated color scheme. Then
they measure to see how much people buy.
IS
College Students and Depression: A public health official is studying differences in
depression among students at two different universities. They collect a random sample
OM
of students independently from each of the two universities and administer a well known
depression inventory. A score of 5 or above indicates some depression. A score above
15 indicates that active treatment is necessary.
Sample Statistics Pictured:
NA
JP
The official conducts a two-sample t-test to determine whether these data provide
significant evidence that students at University 1 are more depressed than students at
University 2. The test statistic is t = 2.64 with a P-value 0.005.Which of the following is
an appropriate conclusion?
