Question

A university Research Group wish to explore whether there are differences in the typical profile of customers who frequent different coffee outlets on the university campus.

A sample of customers were asked which of the four campus coffee outlets they frequent the most, recorded as one of A, B, C, or D. They were also asked to record their age and their monthly expendable income in euro.

The recorded ages for the sample are displayed in the figure below, where a separate boxplot is drawn for the observations from each of the four coffee outlets. Descriptive statistics of age for consumers within each coffee outlet are also provided.

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Descriptives:

Age (years)

Outlet:	Mean:        Standard                                                                                                                          -                 Deviation:
A B C D	21.25          3.11 25.72          5.65 37.94           6.01 43.15           9.33

(a) Identify each of the following variables as either qualitative or quantitative.

(i) which of the four campus coffee outlets a customer frequents the most

(ii) age of customer

(iii) monthly expendable income of customer

(b) The recorded ages for the sample of 8 customers who frequent outlet A are as follows:

17, 18, 20, 21, 21, 22, 25, 26.

(i) Verify that the sample mean for outlet A customers is x = 21:25, as reported in the

table, showing all workings.

(ii) Verify that the sample standard deviation for outlet A customers is s = 3:11, as

reported in the table, showing all workings.

(c) Describe and compare the distributions of age for each of the different coffee outlets.

Include comments about central tendency, spread and the shape of the distributions,

referencing the boxplots and the descriptive statistics provided.

(d) The correlation coefficient between age and monthly expendable income for this sample

was calculated to be r = 0:92. Interpret the meaning of this value in terms of this

application.

Answer 1

Solutionai)

(i) which of the four campus coffee outlets a customer frequents the most

it is qualitative as it has 4 levels(A,B,C,D)

and we cannot measure it

(ii) age of customer

it is quantitative as we can measure it

(iii) monthly expendable income of customer

it is quantitative

(i) Verify that the sample mean for outlet A customers is x = 21:25, as reported in the

table, showing all workings.

mean =xbar=sum of values/total values

=17+18+20+ 21+21+ 22+25+26/

=170/8

=21.25

(verified)

(ii) Verify that the sample standard deviation for outlet A customers is s = 3:11, as

reported in the table, showing all workings.

Sample standard deviation

s= $s=\sqrt{\sum (x-xbar)^2)/n-1}$

X	XBAR	X-XBAR	(X-XBAR)^2
17	21.25	-4.25	18.0625
18	21.25	-3.25	10.5625
20	21.25	-1.25	1.5625
21	21.25	-0.25	0.0625
21	21.25	-0.25	0.0625
22	21.25	0.75	0.5625
25	21.25	3.75	14.0625
26	21.25	4.75	22.5625
		TOTAL	67.5
XBAR=21.25

s=sqrt(67.5/8-1)

=3.105295

s=3.11

(verified)

(c) Describe and compare the distributions of age for each of the different coffee outlets.

Include comments about central tendency, spread and the shape of the distributions,

referencing the boxplots and the descriptive statistics provided.

mean for D is highest

and mean for A is lowest

spread is high for D as standard deviation is high

and spread is least for A.

we can see visually  that 4 groups means are equal

mean of B,C are comparable

spreads are also equal

(d) The correlation coefficient between age and monthly expendable income for this sample

was calculated to be r = 0:92.

means there exists a strong positive relationship between age and monthly expendable income .

As age increases ,monthly expendable income increases and

As age decreases ,monthly expendable income decreases.