Download Homework 6

PubH 6414 Fall2011 Homework 6 (20 points)
We encourage you to work together in computing and discussing the problems.
However, each student is expected to independently write up the submitted
assignment using her or his own computing and giving explanations in her or his
own words. Identical or nearly identical homework submissions will not receive
credit.
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Turn in this completed Word document in class by the homework due date.
You may use R commander to do the calculations needed for each question. Paste in ONLY the
parts of the output needed to answer the question. (You may use another statistical software
package to do the calculations, if you prefer, but the instructor and TAs cannot provide
assistance with other packages.)
Data needed for this homework assignment are on the website link:
http://www.biostat.umn.edu/~susant/FALL11PH6414HMK.html
Problem 1: Multiple Choice Questions. (4 points (0.25 pts. each))
Part A. Suppose that the weights of chicken eggs are normally distributed with population mean 65
grams and standard deviation 5 grams. Suppose that we repeatedly draw a random sample of 12 eggs,
weigh each egg, and calculate the sample mean weight, x , and the sample standard deviation, s.
A1. What is the sampling distribution of the sample mean weights?
a. Normal
b. Standard normal
c. t distribution
d. Binomial
e. Poisson
f. Need more information
A2. What is the mean of the sampling distribution of sample mean weights?
a. 65
b. 65/12
c. 65 / 12
d. 5
e. 5/12
f. 5 / 12
g. Need more information
A3. What is the standard error (SE) of the sampling distribution of sample mean weights?
a. s
b. s/12
c. s / 12
d. 5
e. 5/12
f. 5 / 12
g. Need more information
A4. As sample size increases, what happens to the mean of the sampling distribution?
a. It increases
b. It decreases
c. It stays the same
d. Need more information
A5. As sample size increases, what happens to the SE of the sampling distribution?
a. It increases
b. It decreases
c. It stays the same
d. Need more information
Part B. Suppose that the weights of duck eggs are normally distributed with population mean 80 grams
but unknown standard deviation. Suppose that we repeatedly draw a random sample of 12 eggs, weigh
each egg, and calculate the sample mean weight, x , and the sample standard deviation, s.
B1. What is the sampling distribution of the sample mean weights?
a. Normal
b. Standard normal
c. t distribution
d. Binomial
e. Poisson
f. Need more information
B2. What is the mean of the sampling distribution of sample mean weights?
a. 80
b. 80/12
c. 80 / 12
d. 5
e. 5/12
f. 5 / 12
g. Need more information
B3. What is the standard error (SE) of the sampling distribution of sample mean weights?
a. s
b. s/12
c. s / 12
d. 5
e. 5/12
f. 5 / 12
g. Need more information
B4. As sample size increases, what happens to the mean of the sampling distribution?
a. It increases
b. It decreases
c. It stays the same
d. Need more information
B5. As sample size increases, what happens to the SE of the sampling distribution?
a. It increases
b. It decreases
c. It stays the same
d. Need more information
Part C. Suppose you are a researcher interested in obesity. Suppose that you know that body mass
index (BMI) in the population you are interested in is normally distributed with mean  and standard
deviation . Suppose that you send each of your five student research assistants out to take a random
sample of 40 people from this population, measure each person's BMI, and calculate the sample mean
BMI, x , and the sample standard deviation, s, for their 40 people.
C1. The mean BMI in the population is .
a. True
b. False
c. It depends
C2. Each student's sample will have the same sample mean BMI as every other student's
sample.
a. True
b. False
c. It depends
C3. Each student's sample will have the same sample standard deviation, s, as every other
student's sample.
a. True
b. False
c. It depends
C4. The mean of the sampling distribution of sample means will equal .
a. True
b. False
c. It depends
C5. The standard error (SE) of the sampling distribution of sample means will equal .
a. True
b. False
c. It depends
C6. Some of the students' samples will have sample means, x , that are larger than the
population mean, .
a. True
b. False
c. It depends
Problem 2. Serum Sodium Levels Revisited (6 points (1 pt. each))
The values of serum sodium in healthy adults approximately follow a normal distribution with a mean
= 141 mEq/L and a known standard deviation () = 3 mEq/L.
Use the pnorm function (or R Commander menu options) to answer the following questions. Please
give the pnorm formula you used (or the formula R Commander used) as well as the result.
A. What is the probability that the mean serum sodium in a random sample of 36 healthy adults
is < 140?
B. What is the probability that the mean serum sodium in a random sample of 16 healthy adults
is < 140?
C. Compare your answers to parts A and B. Are they different? Explain why or why not.
D. What is the probability that the mean serum sodium in a random sample of 36 healthy adults
is between 140 and 142?
E. What is the probability that a healthy adult will have a serum sodium value between 140 and
142?
F. Compare your answers to parts D and E. Are they different? Explain why or why not.
Problem 3. Birth Weights Revisited (4 points (1 pt. each))
A cohort study was carried out involving 18,665 Caucasian infants born at Montreal's Royal Victoria
Hospital from January 1978 to March 1990. The birth weights of those infants were normally
distributed with a mean =3.369 kg and standard deviation = 0.567 kg. (Reference: Shi Wu Wen,
Michael S. Kramer, and Robert H. Usher, Comparison of Birth Weight Distributions between Chinese
and Caucasian Infants, Am. J. Epidemiol. 1995; 141: 1177-1187.)
Suppose the investigators randomly sampled 100 infants out of the 18,665 infants for a new study.
A. What are the mean and standard error (SE) of the sampling distribution?
B. What is the probability that the mean birth weight of this sample of 100 infants is less than
3.2 kg? Please give the pnorm formula you used (or the formula R Commander used) as well as
the result.
C. What is the probability that the mean birth weight of these 100 infants is more than 3.5 kg?
D. What is the probability that this sample of 100 infants will have a mean weight between 2.5
kg and 3.5 kg?
Challenge Problem (2 points extra credit)
E. Suppose that for a new study, the investigators need to determine the size of a sample of
infants, such that 95% of the means of the samples of this size will be less than 3.5 kg. What is
the sample size needed? (Hint: Calculate the z-value first.) (Note: Sample sizes should be
rounded up to the nearest integer.) [Please show your work.]
Problem 4. ACT Test Scores (6 points (1 pt. each))
According to www.act.org, for the American high school graduating class of 2001, the national
average composite score on the ACT exam was 21.0, with a standard deviation of 4.7. We can
reasonably assume that ACT scores in this national population are normally distributed.
Please show your work, including R or Rcmdr formulas you used, for all questions.
A. What is the probability that a randomly selected 2001 graduate had an ACT score between 17
and 25?
B. What is the 90th percentile of 2001 ACT scores?
C. How unusual is a composite ACT score of 22 or better, that is, what is the probability that a
randomly selected 2001 graduate had a score above 22?
D. What is the probability that the average ACT score for a random sample of 100 high school
graduates would be between 17 and 25?
E. What is the 90th percentile of the average ACT scores for random samples of 100 high school
graduates?
F. How unusual would it be to find a random sample of 100 high school graduates where the
sample average ACT score was above 22?