Assignment4.pdf - COR-STAT 1203 Introduction to Statistical...

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COR-STAT 1203 Introduction to Statistical Theory, G3-G4, Term II, 2021-22
Assignment 4
Due date and time: Friday 11.59pm, Week 11 (March 25, 2022)
Place to submit: eLearn → Assignments → Assignment 4
Instructions:
(i) Write your name and group number clearly on the top of the first page;
(ii) Convert your solutions into one pdf file named: YourName_A4_GroupNo.pdf ,
e.g., GohZhiHui_A4_G3.pdf, TayAlbert_A4_G4.pdf;
(iii) Late assignment will not be accepted, and zero mark will be given to two or more
very similar assignments;
(iv) You may do the assignment by handwriting on A4-size paper → scan → pdf;
or typing by word (or some other software, with A4 page specification) → pdf;
(v) Minimize the size of the pdf file for submission.
1. A random sample X
1
, X
2
, …, X
n
of size n is taken from an exponential distribution
with mean β . Let X be the sample mean.
(a) Show that 𝑋𝑋
�
is an unbiased estimator of β
(b) Show that Var
(
𝑋𝑋
�
) =
𝛽𝛽
2
𝑛𝑛
⁄
(c) Based on a random sample of size n = 50, 𝑋𝑋
�
is observed to be 12. Give an
approximate 95% confidence interval for β
2. The heights of a random sample of 50 college students showed a mean of 174.5
centimeters and a standard deviation of 6.9 centimeters.
(a) Construct a 95% confidence interval (C.I.) for the mean height of all college
students.
(b) What can we assert with 95% confidence about the possible size of our error
if we estimate the mean height of all college students to be 174.5 centimeters?
(c) Construct a 99% C.I. for the mean height of all college students. Compare
with the 95% C.I. obtained in (a) and comment.
3. The following measurements were recorded for the drying time, in hours, of a
certain brand of latex paint:
3.4, 2.5, 4.8, 2.9, 3.6, 2.8, 3.3, 5.6, 3.7, 2.8, 4.4, 4.0, 5.2, 3.0, 4.8.
(a) Provide point estimates for the population mean and standard deviation of the
drying time.
(b) Assuming the measurements represent a random sample from a normal
population, find a 90% confidence interval and a 95% confidence interval for
the population mean drying time.
(c) Suppose that it is known (from previous studies) that the population standard
deviation of drying time σ = 1. How large a sample will be needed in order for
the sample mean to be within 0.2 hour of the true mean with 95% confidence?
.
.
.

COR-STAT1203 A4