Answer to Question #148726 in Statistics and Probability for Daphne

Question

(a) The supermarket statistician realized that there was a considerable range in the spending

power of its customers. Even though the overall spending seemed to have increased the

high spenders still spent more than the low spenders and that the individual increases would

show a smaller spread. In other words these two populations, ‘before’ and ‘after’, and not

independent.

Before the next advertising campaign at the supermarket, he took a random sample of 10

customers, Α to J , and collected their till slips. After the campaign, slips from the same

10 customers were collected and both sets of data recorded. Using the paired data, has there

been any mean change at a 95% confidence level?

A B C D E F G H I J

Before 42.30 55.76 32.29 10.23 15.79 46.50 32.20 78.65 32.20 15.90

After 43.09 59.20 31.76 20.78 19.50 50.67 37.32 77.80 37.39 17.24

Answer 1

Solution

Let X-Y=D

$\sum D=-32.93$

$\sum D^2=210.8627$

$H_0: \mu_1 - \mu _2=0$ vs

$H_1: \mu_1 - \mu_2 \not= 0$

Test statistic :

$$t-stat= {{(\sum D) \over N} \over \sqrt {{\sum D^2}-{{(\sum D) } ^2 \over N} \over N(N-1)}}$$

$$={ {- 32.93 \over 10} \over \sqrt {210.8627-{{(-32.93)}^2 \over 10} \over 10 *9} }$$ $$=-3.0868$$

$t_{0.025,9}=2.262 \space and \space - 2.262$

Since the test statistic does not lie between 2.262 and - 2.262, it lies in the critical region hence we reject the null hypothesis that there is no difference between the means.

Therefore we conclude that there is a difference between the spending power of customers before and after the advertising campaign