Probability and Statistics – Mrs. Leahy
Unit 8 Homework Problems
Section 1: Estimating µ when σ is known
COMPLETE THE FOLLOWING ON YOUR OWN PAPER
1. Allen’s hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther. A small group of 15 Allen’s
hummingbirds has been under study in Arizona. The average weight of these birds is 𝑥̅ = 3.15 grams. Based on
previous studies, we can assume that the weight of Allen’s hummingbirds have a normal distribution, with σ =0.33
grams.
a) Find an 80% confidence interval for the average weights of Allen’s hummingbirds in the study region.
What is the margin of error (E)?
b) What conditions are necessary for your results?
“The x distribution is normal” or “The sample size is ≥ 30”
c) Give an interpretation of your results. We can conclude that…. etc
2. Total plasma volume is important in determining the required plasma component in blood replacement
therapy for a person undergoing surgery. Plasma volume is influenced by the overall health and physical activity
of an individual. Suppose that a random sample of 45 male firefighters are tested and that they have a plasma
volume sample mean of 𝑥̅ = 37.5 𝑚𝑙/𝑘𝑔 . Assume that σ=7.50 ml/kg for the distribution of blood plasma.
a) Find a 99% confidence interval for the population mean blood plasma volume in male firefighters.
What is the margin of error (E)?
b) What conditions are necessary for your results?
“The x distribution is normal” or “The sample size is ≥ 30”
c) Give an interpretation of your results. We can conclude that…. etc
3. Thirty small communities in Connecticut (population near 10,000 each) gave an average of 𝑥̅ = 138.5
reported cases of larceny per year. Assume that σ is known to be 42.6 cases per year.
a) Find a 90% confidence interval for the population mean annual number of reported larceny cases in
such communities. What is the margin of error?
b) Find a 95% confidence interval for the population mean annual number of reported larceny cases in
such communities. What is the margin of error?
c) Find a 99% confidence interval for the population mean annual number of reported larceny cases in
such communities. What is the margin of error?
d) Compare the margins of error for parts (a) to (c). As the confidence levels increase, what happens to
the margins of errors?
e) Compare the LENGTHS of the confidence intervals for parts (a) to (c). As the confidence levels
increase, what happens to the length of the confidence intervals?
Answer TRUE or FALSE:
4. The value zc is a value from the standard normal distribution such that P(  zc  z  zc )  c .
5. The point estimate for the population mean µ of an x distribution is 𝑥̅ , computed from a random
sample of the x distribution.
6. Consider a random sample of size n from an x distribution. For such a sample, the margin of error for
estimating µ is the magnitude of the difference between 𝑥̅ and µ.
7. Every random sample of the same size from a given population will produce the exact same confidence
interval for µ.
8. A larger sample size produces a longer confidents interval for µ.
9. If the standard deviation for an x distribution decreases, c confidence intervals based on the same
sample size will become shorter.
10. If the sample mean 𝑥̅ of a random sample from an x distribution is relatively small, then the
confidence interval for µ will be relatively short.
11. For the same random sample, when the confidence level c is reduced, the confidence interval for µ
becomes shorter.
Section 2: Estimating µ when σ is unknown
COMPLETE THE FOLLOWING ON YOUR OWN PAPER
Note: Use the information from part a), but you do not have to verify the information using a calculator.
1.
2.
3.
Section 3: ESTIMATING p
Use your yellow textbook to do the following
pg 456-457 #11,12,16, 17, 23, 34