Exam III
1. (5 points) Define "Sampling Distribution".
The pdf of a sample statistic.
2. (5 points) Define "power" of a statistical hypothesis
test.
The probability of rejecting the null hypothesis when the null hypothesis is false.
3. (5 points) The sample median (m) is an unbiased estimator of the
population mean (m) when the sampled population (the parent population)
is symmetric. What does the phrase "the sample median is an
unbiased estimator of the population mean" mean?
The mean of the sampling distribution of the sample median is the
population mean (mean of the parent population).
4. (5 points) State the Central Limit Theorem.
For large enough n, 
is approximately normally distributed with mean
and standard
deviation 
when the sample was collected randomly from any population with mean and standard deviation 
5. (5 points) What is the difference between the sample standard
deviation and the sample standard error? What does each estimate?
The sample standard deviation is used to estimate the variability of
the parent population. The sample standard error is used to estimate
the variability of the sampling distribution.
6. (10 points) What is the difference between Type I and Type II
errors in hypothesis testing? How do 
and 
relate to Type I and Type II errors?
Type I errors are when you reject the null falsely (meaning that you
reject the null when it is true). Type II errors are when you fail to
reject the null when it is true.
is the probability of a Type I error and
is the probability of a Type II error.
7. (10 points) Find P(7.6 
8.8) when
= 10, s2 = 36 and n = 25.
P(7.6
8.8) =
P[(7.6 - 10)/sqrt(36/25) z (8.8 - 10)/sqrt(36/25)] =
P(-2 z -1) =
0.1587 - 0.0228 =
0.1359
8. (10 points) Find P(
> 0.55) when p = 0.5 and n=500.
P( >
0.55) =
P{z > (0.55 - 0.5)/sqrt[0.5*(1-0.5)/500]} =
P(z > 2.24) =
0.0125
9. A random sample of 16 subjects resulted in a sample mean of
98 and sample standard variance of s2 = 160.
a. (10 points) Find a 95% confidence interval for the population
mean (m) based on this data.
98 
2.131*sqrt(160)/sqrt(16) =
98
6.74
(91.26, 104.74)
b. (5 points) When we interpret what a confidence interval is we
say that "We are 95% confident that the population mean falls within the confidence interval. Fill in the
blanks (two words).
c. (5 points) What do we mean by the phrase "95% confident"
in the above interpretation?
If we were to take lots of random samples from the population and
calculated a 95% confidence interval from each, then 95% of them
would include the population mean in that interval
10. (10 points) In the Lovastatin study the treatment group had a
mean reduction in cholesterol levels of -20% where s2 = 260 and n =
65. Calculate a 90% confidence interval for m, the population mean
reduction of cholesterol levels.
-20
1.645*sqrt(260)/sqrt(65) =
-20
3.29
(-23.29, -16.71)
11. Mark each of the following statements as True or False by
circling the correct answer. Justify your answer on the back of the page
if you feel you need to. (3 points each)
a. If you increase alpha then beta will automatically decrease. True
b. If you increase the sample size then alpha will automatically
decrease. False
c. If you increase the sample size then Power of the hypothesis test will automatically increase.
True
d. When the confidence level of a confidence interval is increase the
width of the confidence interval will decrease. False
e. When sample size increases the width of a confidence interval will tend to increase.
False
12. (10 points) The sampling distribution of is the normal
distribution (as long as it's from a random sample) when: (circle all
that apply)
a. The parent population is normal and the sample is
"large"
b. The parent population is not normal and the sample is
"large"
c. The parent population is normal and the sample is
"small"
d. The parent population is not normal and the sample is
"small"
e. The sampling distribution is always normal
Answers: a, b and c
