Question

Your friend owns a retail store where she plans to give arriving customers a short verbal ”commercial” message regarding products for sale. Your friend asks you to determine if this will boost the average dollar value of sales to each customer, µ, which is currently $5.00. You may assume that the standard deviation per transaction is $1.00 and that the population is of unlimited size. You monitor a random sample of n = 200 customers given the short verbal ”commercial” message upon arrival at the store and determine the amount purchased by each. (a) Your friend will only adopt the policy of giving short verbal ”commercial” messages to each newly arriving customer if the sample indicates that the values of sales per customer will improve. Formulate the null and alternative hypotheses. [2 marks] (b) Suppose you wish to test the null hypothesis while ensuring that your friend only has a 5% chance of getting a recommendation to adopt the policy of giving short verbal ”commercial” messages when that policy actually does not improve the value of sales per customer. Describe the test statistic you would use to conduct the test, compute the critical region for the test, and describe the decision rule you would use. [4 marks] (c) You collect the sample data. What recommendation would you make to your friend if (i) X¯=$5.37, (ii) X¯=$5.05, (iii) X¯=$4.97, and (iv) X¯=$5.20 ? Express your recommendations as formal hypothesis testing conclusions. [4 marks]

Answer 1

Given population mean = $5.00

population std dev = $1.00

a) We to test the claim that the values of sales per customer will improve.

Hypothesis

H0 : $5.00

H1 : $5.00

b) Describe the test statistic you would use to conduct the test?

Since the population parameter is known, and we have only one sample. Hence we perform one sample Z-test.

test statistic Z =

given sample size n = 200.

level of significance alpha = 0.05

critical region:- Z > Z0.05 (1.645) because it is a right tailed test.

decision rule:- If test statistic is more than 1.645 then reject H0 otherwise fail to reject H0..

c)

(i) X¯=$5.37

Z = = 5.233

Z > Z0.05, we reject H0, there is a significant evidence to conclude that giving the short verbal ”commercial” message upon arrival at the store increases the amount purchased.

(ii) X¯=$5.05,

Z = = 0.707

Z < Z0.05, we fail to reject H0, there is no significant evidence to conclude that giving the short verbal ”commercial” message upon arrival at the store increases the amount purchased.

(iii) X¯=$4.97,

Z = = -0.424

Z < Z0.05, we fail to reject H0, there is no significant evidence to conclude that giving the short verbal ”commercial” message upon arrival at the store increases the amount purchased.

(iv) X¯=$5.20

Z = = 2.828

Z > Z0.05, we reject H0, there is a significant evidence to conclude that giving the short verbal ”commercial” message upon arrival at the store increases the amount purchased.