# STATISTICS CALCULATION

QUESTION

200032 Statistics for Business
Assignment 2

Due date: Wednesday, 23
24.
rd
May

Value: 25%

This question sheet provides the basic outline of each assignment question. The data for this assignment
will be supplied separately with each student receiving different data. Your data set number should be
written clearly at the beginning of your completed assignment.

You should complete and sign a cover sheet (see above) and attach it as the FIRST page of your completed
assignment.

In answering these questions, it is YOUR comments which are required. The commentary and
interpretation parts of each question carry a reasonably high proportion of the marks.

You should produce any graphs by using a computer graphics or spreadsheet package such as
Microsoft Excel.

In the hypothesis testing parts of questions 1 to 5, your answer should include:

A. the (assumed) distribution of the data (where appropriate);
B. the hypotheses being tested and relevant parameter values;
C. the calculation of a suitable test statistic;
D. a graph showing the critical region for the test;
E. an indication of whether the null hypothesis is accepted or rejected and why;
F. a suitable conclusion.

You should assume that all the populations being studied are large.
QUESTION 1:

A recent survey of _____ guests at the Backpackers Paradise Hotel gave a sample average daily food
x
= \$_____ with a sample standard deviation, s = \$_____.

expenditure of
a. Calculate a 95% confidence interval for the unknown population mean, μ.

b. What sample size would be required to be 95% sure of estimating the unknown population mean to
within ±\$5?

c. A census of guests 2 years ago gave a population mean expenditure of \$_____. Test, at the 5% level
of significance, whether your data supports the hypothesis of a significant increase in the average daily
food expenditure over the 2-year period.
[Hint: This is a 1-sample test]
QUESTION 2:

Magoos is a large employer interested in comparing the days lost due to illness or injury for its male and
female workers. Random samples of male and female workers gave the following summary statistics for the
number of days lost last year:

Sample size (n)
Average (
x
)
Standard Deviation (s
)
Males

Females

[In the parts of the question below, you may assume that the unknown population standard
deviations for males and females are equal]

a. Calculate a 99% Confidence Interval for the difference in the population means.

b. Test, at the 1% level of significance, whether the population means could be equal against an alternative
that the population means are different for females compared to males.

c. Show that the results in a. and b. are consistent. That is, if zero falls into the confidence interval in a.,
then the null hypothesis in b. is accepted OR if zero does not fall into the confidence interval in a., then
the null hypothesis in b. is rejected.

QUESTION 3:

You are undertaking a comparative study of the ownership of commercial premises in two locations.
At location A, you find that of premises surveyed, are owner occupied while at location B you
find that of premises surveyed, are owner occupied.

a. Construct a 95% confidence interval for the difference in the population proportions.

b. Is this data sufficient to conclude, at the 5% level of significance, that location A has a significantly higher
proportion of owner-occupiers compared to location B?

25.
x
Introductory Book
200032 Statistics for Business

QUESTION 4:

In an attempt to examine the historical difference between two investments, the return from each investment
was recorded over the last 10 reporting periods. The results were:

26.
Program Rate of Return (%)
Period Investment 1 Investment 2
1

2

.

.

9

10
Is this a paired test? Why or why not?

Assuming the investment returns are normally distributed, can we conclude that the average rate of
return for Investment 1 is greater than the average return for Investment 2 at the 5% level of
significance?

QUESTION 5:

A weight-loss company claims that its weight-loss program is superior to its competitor. To test this claim, a
group of 20 overweight male volunteers were randomly allocated to one program or the other and the weight
losses over a 3-month period were recorded:

Weight Loss (kgs)
Company Program Competitor’s Program

Use a suitable parametric test, with a 1% level of significance, to assess the company’s claim. Why will this
be a 1-sided test? [Note that the data is not paired (matched) in this question]

QUESTION 6:

The following data gives the frontage (in metres) and annual rent (in \$) of a number of retail shops in a suburban
shopping mall:

Frontage (m) Annual Rent (\$)

a. Using Excel, graph the data (Annual Rent vs Frontage). How well does a straight line relationship fit
the data?
b. Use simple linear regression to develop the estimated regression line that relates annual rental to
frontage.
c. Calculate a 95% confidence interval for the true, unknown slope parameter, β.
d. What is the correlation coefficient between frontage and annual rent? How strong is the relationship
between frontage and annual rent for this set of data?
e. What does the model predict to be the approximate increase in the Annual Rental for each one
metre increase in the Frontage (ie interpret the slope estimate)?
f. Use your regression equation to estimate the rental for a comparable premises in the mall with a
frontage of metres. How confident are you in this estimate? Why?
g. A potential tenant in the mall feels that she can only afford to pay \$ in annual rent for her
premises. Use the fitted model to estimate approximately what length of shop frontage she could
obtain.
h. How confident are you in the predictions in f. and g.? Comment briefly.
i. Carry out the test (at a 1% level of significance) that the true, unknown slope of the regression line is
zero against an alternative that the slope is different from zero. (ie H0: β = 0; HA: β ≠ 0 or H0: ρ = 0;
HA: ρ ≠ 0). Is there a significant linear relationship?

[IT IS RECOMMENDED THAT YOU DO THE CALCULATIONS USING YOUR CALCULATOR
AND CHECK THEM ON EXCEL – answers to c. and i. are obtainable from your Excel printout

27.
Introductory Book

SOLUTION

6

a)      This is the chart of Frontage (in m) with the rent (in\$). This is clearly a linear relation-ship as shown by the linear tread line.

b)      Annual Rent = 3139.818 +2835.748*(Frontage)

c)      The slope parameter for 95% confidence parameter is 4085.058836.

d)      The correlation coefficient between frontage and the rent is 0.848.The strength of the relationship is 0.719.

e)      The linear regression model is best suitable for the data with slope 2835.748.

f)       When F=5.3m Rent=15784.2824 from relation in (b)

g)      Rent=25300, Frontage= 7.81 m fro relation in (b)

h)      The relation in f and g are confident to approximately 72%

i)       For 1% level of significance:

 j) k)      Upper 99% Lower 99% Intercept -6794.716876 13074.35238 Slope 1058.74682 4612.749671

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