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ENGRD 2700: Basic Engineering Probability and Statistics
Fall 2023
Homework 7
Due Wednesday,
November 8th
at 11:59pm. Submit Solutions to Gradescope by clicking the name of the
assignment. See syllabus for detailed submission instructions.
When completing this assignment (and all subsequent ones), keep in mind the following:
•
You must complete the homework individually and independently.
•
Provide evidence for each of your answers. If a calculation involves only very minor computation then
explain the computation you performed and give the results. If a calculation involves more complicated
steps on many many records then hand in the calculations and formulas for the first few records only.
•
Write clearly and legibly. You are encouraged to
type
your work although you do not have to. We may
deduct points if your answers are di
ffi
cult to read or disorganized.
•
For questions that you answer using Python, attach any code that you write, along with the relevant
plots.
•
Submit your homework a single pdf file on Gradescope.
1. A sample of 100 service times at a call center has a sample mean of 9 minutes and a sample standard
deviation of 6 minutes. Assume that the service times are independent and have a normal distribution.
(a) Give a 95% confidence interval for the mean service time.
(b) Approximately how many service times we would have to collect to return a 95% confidence
interval whose width is at most 15 seconds (= 1/4 minutes)?
(c) We want to quote a time
T
to a customer so that, with 95% probability, the (random) service
time will be less than
T
. Does your answer in Part (a) help? If yes, how? If no, explain why not.
2. Ratan Tata and Pete Coors are running for state governor, and two well known Super-PACs, Sega-
Genesis and Super-Nintendo, decide to gauge public opinion.
(a) Super-Sega-Genesis interviews 453 people, and finds that 55% of individuals want to vote for
Ratan. Construct a 95% confidence interval for
p
, the proportion of all voters in the state sup-
porting Ratan.
(b) Super-Nintendo conducts its own independent study, and obtains the interval [0
.
492
,
0
.
568] from
a sample size of 378. What confidence level did Super-Nintendo use?
(c) Suppose, contrary to the information specified in part (b), that Super-Nintendo obtained the 90%
confidence interval [0
.
509
,
0
.
591] instead. How many individuals did Super-Nintendo interview?
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3. Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is
normally distributed with true standard deviation 0
.
75.
(a) Compute a 95% confidence interval for the average porosity of a certain seam if the average
porosity for 20 specimens from the seam was 4
.
85.
(b) Compute a 98% confidence interval for the true average porosity of another seam based on 16
specimens with a sample average porosity of 4
.
56.
(c) How large a sample size is necessary if the width of the 95% interval is to be 0
.
4?
4. Mighty Quinn (my favorite bbq joint in NYC) is known far and wide for his brisket sandwiches.
Although Mighty Quinn advertises that the sandwiches weigh 400 g, Bianca Wu (Jazz aficionado), a
regular, suspects that the sandwiches weigh less. Bianca buys a sandwich each day for 81 consecutive
days. She obtains the 95% confidence interval [393
.
08
,
400
.
92] for the mean weight
μ
of a sandwich.
(a) Find ¯
x
and
s
, the mean and standard deviation of her sample.
(b) Approximately how many sandwiches would Bianca need to buy in order to halve the width of
her confidence interval?
(c) Compute 90% and 99% confidence intervals for
μ
.