STAT 3115 – Homework #9

STAT 3115 – Homework #9
Due Date: Thursday, April 20th, Beginning of Class
Note: Please include the following information on the first page of your completed homework.
1. Your Name
2. STAT 3115 – Section: 1C
3. Homework #9
4. Due Date: Thursday, April 20th
5. List of students you worked with on this assignment (if applicable)
Guidelines: Please remember to show your work and explain your answers as necessary. Answers that
are not supported by reasoning will not receive full credit. Homework should be stapled if it is longer
than one page. Numbered problems are from Devore, 9th ed. unless otherwise noted.
Section 5.4:
1- Ice cream manufacturing. Ice cream cartons from a certain manufacturer have a true average
weight of 12 ounces, with a population standard deviation of 1 ounce. You take a random sample
of 100 cartons.
a. Find the expected value and standard deviation of the average weight (?????) in the sample of 100
b. Find the (approximate) probability that the average weight among these 100 cartons is between
11.8 and 12.15 ounces.
c. What theorem are you using to calculate the probability in b? What assumption is necessary in
order for this theorem to hold?
2- Young’s modulus is a quantitative measure of stiffness of an elastic material. Suppose that for
aluminum alloy sheets of a particular type, its mean value and standard deviation are 70 GPa and
1.6 GPa, respectively.
a. If ????? is the sample mean Young’s modulus for a random sample of ???? = 16 sheets, where is the
sampling distribution of ????? centered, and what is the standard deviation of the ????? distribution?
b. Answer the questions posed in part (a) for a sample size of ???? = 64 sheets.
c. For which of the two random samples, the one of part (a) or the one of part (b), is ????? more likely
to be within 1 GPa of 70 GPa? Explain your reasoning.
Now suppose the distribution is normal (the cited article makes that assumption and even
includes the corresponding normal density curve).
d. Calculate ????(69 = ????? = 71) when ???? = 16.
e. How likely is it that the sample mean diameter exceeds 71 when ???? = 25?
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Section 7.1:
3- Helium porosity. Assume that the helium porosity (in percentage) of coal samples taken from a
particular seam is normally distributed with a population standard deviation of 0.69. We are
interested in constructing a confidence interval for the true average helium porosity (µ) in the seam.
a. Find a 95% confidence interval for µ when a random sample of 25 coal samples are taken
yielding a sample mean of 4.56.
b. Find a 95% confidence interval for µ when a random sample of 100 coal samples are taken
yielding a sample mean of 4.56.
c. Compare the widths of a and b, stating how the sample size affects the width.
d. Find a 98% confidence interval for µ when a random sample of 100 coal samples are taken
yielding a sample mean of 4.56.
e. Compare the widths of b and d, stating how the confidence level affects the width.
f. How large of a sample size is needed to ensure a width of 0.4 for a 95% confidence
4- Each of the following is a confidence interval for ???? = true average (i.e., population mean)
resonance frequency (Hz) for all tennis rackets of a certain type: (114.4, 115.6) (114.1, 115.9)
a. What is the value of the sample mean resonance frequency?
b. Both intervals were calculated from the same sample data. The confidence level for one of
these intervals is 90% and for the other is 99%. Which of the intervals has the 90% confidence
level, and why?
Section 7.2:
5- Night School Students. A sample of 60 night school students is obtained in order to estimate the
mean age of all night school students at a university. The result is a sample mean of 25.3 years and
a sample standard deviation of 4.
a. Give a point estimate for µ, the population mean age of all night school students.
b. Find a 99% confidence interval for µ.
6- A sample of 56 research cotton samples resulted in a sample average percentage elongation of 8.17
and a sample standard deviation of 1.42 (“An Apparent Relation Between the Spiral Angle f, the
Percent Elongation E1, and the Dimensions of the Cotton Fiber,” Textile Research J., 1978: 407–
410). Calculate a 95% large-sample CI for the true average percentage elongation m. What
assumptions are you making about the distribution of percentage elongation?
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