Stat 2606: business statistics i – assignment 4

Carleton University
School of Mathematics and Statistics
STAT 2606: Business Statistics I – Assignment 4
Section C due Tuesday, November 17 at 2:25pm.
Section D due Wednesday, November 18 at 2:25pm.
INSTRUCTIONS:
I) Assignments are to be submitted in-class on the due date and prior to beginning of the lecture. No
late assignments will be accepted without sufficient advanced noticed and a legitimate,
documented reason.
II) You must show all of your work. No credit will be given for answers without justification. You must
explicitly define any random variable you use in a solution. No credit will be given for illegible work.
III) Do not use MINITAB for any part of a question unless it specifically says to do so. For questions
that require MINITAB, you must include all relevant output with your assignment.
IV) The lab for this assignment will take place during the week of November 9.
V) This assignment is intended to represent your individual knowledge. It is not a group assignment.
Part A MINITAB QUESTIONS
Question 1. [Central limit theorem (CLT)] First, enable commands in the SESSION window as
follows:
CLICK Editor, then CLICK Enable Commands
You should now see an MTB > prompt in the session window. If so, you are ready to begin typing,
into the SESSION window, the commands in the boxes below.
Generate and store, in columns c3-c902, a total of 100 row-wise samples, each of size n = 900, from
a Poisson distribution with parameter λ = 9 as follows (Note that there is a semi-colon at the end of the
first line, and a period at the end of the second):
random 100 c3-c902;
poisson 9.
1
Note This may take a few moments as you are generating 900 × 100 = 90,000 values.
Create and store in column c1 the 100 values of X based on the 100 samples of same size n = 900 as
follows:
rmean c3-c902 c1
(a) Print (and include in your assignment) the boxplot of c3. According to the position of the median,
what can you conclude about the shape of this data set?
boxplot c3
(b) Use the describe command to find the sample mean and standard deviation of c3 by typing the
following command in the SESSION window:
desc c3
(c) Print (and include in your assignment) the boxplot for the data in column c1. What can you conclude
about the shape of data in c1?
boxplot c1
(d) Use the describe command to find the sample mean and standard deviation of c1 by typing the
following command in the SESSION window:
desc c1
Are they close to 9 and 3/30? Why?
Before starting question 2, you should open a new MINITAB Worksheet(Select File → New
→ Minitab Worksheet) or erase the data from question 1 by typing the following command
in the session window:
erase c1-c1000
Question 2. [Confidence interval (CI) for a mean:] Similar to QUESTION 1, you will be
typing the commands in the boxes below into the SESSION window in MINITAB.
We want to build 100 confidence intervals (CIs) with confidence level (1−α)100% = 95% for the mean
µ of a Normal distribution via the following steps:
Step 1. Generate and store in columns c6-c30 100 samples of size 25 each from Normal with parameter
µ = 2, σ = 5 as follows:
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random 100 c6-c30;
normal 2 5.
Step 2. Use columns c4 and c5 to store respectively the means and the standard deviations of the 100
samples you generated in step 1, as follows:
rmean c6-c30 c4
rstd c6-c30 c5
Step 3. Store the lower bound and the upper bound of your 95% CIs in c2 and c3 respectively by
typing successively:
let c2=c4-1.96*c5/5
let c3=c4+1.96*c5/5
Then create a column c1 containing 1 or 0 according to whether the corresponding interval [c2 , c3]
covers µ or not, by typing:
let c1=(c2 <= 2 and c3 >= 2)
Finally sum up the entries of column c1 to find how many CIs cover the value µ = 4 by typing:
tally c1
(a) What is the percentage of confidence intervals that contain the true value µ = 2?
(b) How do you compare this percentage to the confidence level 95%?
Page 3
Part B Written Questions
Question 1. Suppose a sample of 120 items is drawn from a population of manufactured products and
the number of defective items is recorded. Prior experience has shown that the proportion of defectives is
0.05.
(a) Describe the sampling distribution of b p, the proportion of defectives.
(b) What is the probability that the sample proportion is less than 0.10?
Question 2. Statistics suggest that software developers with two years of experience in Ontario earn an
average of 70,000 per year, with a standard deviation of 5,000. To verify this salary level, a random
sample of 100 software developers with two years of experience was selected from a personnel database for
all software developers in Ontario.
(a) Describe the sampling distribution of the sample mean
¯
X of the average salary of these 100 Software
Developer.
(b) Calculate the probability that X is greater than or equal to 71,200.
(c) If the random sample actually produced a sample mean of 71,200, would you consider this rather
unusual? What conclusion might you draw then?
Question 3. The college president asks the statistics teacher to estimate the average age of the students
at their college. How large a sample is necessary? The statistics teacher would like to be 99% confident
that the estimate should be accurate within 0.5 year. From a previous study, the standard deviation of
the ages is known to be 2 years.
Question 4. Suppose that interest lies in estimating the proportion of drivers who speed on the highway.
What sample size is needed to estimate this proportion to within 5% with 90% confidence?
Question 5. The economy suffered significantly during 2008. There were a large number of foreclosures
of family homes. In one large community, realtors randomly sampled 36 bids from potential buyers to
determine the average loss in home value. The sample showed the average loss was $11,560. Suppose it is
known that σ = $1500.
(a) Find a 80% confidence interval for the mean loss in value per home.
(b) Interpret this interval in (a) and explain what 80% confidence means.
(c) Is the estimate in part (a) valid? Explain.
Question 6. In a survey of 1,000 people, 420 are opposed to the tax increase.
(a) Construct a 95 percent confidence interval for the proportion of those people opposed to the tax
increase.
(b) Interpret the CI in terms of the question.
(c) Is the estimate in part (a) valid? Explain.
Question 7. A sample of 8 items has an average fat content of 18.6 grams and a sample standard deviation
of 2.4 grams. Assuming a normal distribution.
(a) Construct a 99 percent confidence interval for the population mean.
Page 4
(b) Interpret the CI in terms of the question.
(c) Is the estimate in part (a) valid? Explain.
Page 5

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