Question S9: Yummy Superstores is considering updating the checkout operating sy

ID: 3376586 • Letter: Q

Question

Question S9: Yummy Superstores is considering updating the checkout operating system to ensure speedier checkout operations. Tes new system suggests customers will wait 4.2 minutes on average with a standard deviation of ts on customer waiting times provided by the manufacturers of the utes. This is based on an observation of 1000 customers. Yummy Superstores samples tes with their existing systems for comparison and finds customers wait, on average, 3.5 minu a standard deviation of 1.9 minutes. This is based on observations of 500 customers. What recommendation would you make (assuming ?=.05 level of significance? (a) Yummy Superstores should consider adopting the new system as the average (b) Yummy Superstores should not consider adopting the new system as the average (c) The action for Yummy Superstores waiting times with the new system are significantly superior relative to the existing system. waiting times with the new system are not significantly superior relative to the existing system is unclear as the sample sizes for the two systems differed. Yummy Superstores should sample more customers. he action for Yummy Superstores is unclear as we are unsure that the population distribution of checkout times is normal under either system. (d) T

Explanation / Answer

Solution

Answer: Option (a)

Details of working

Let X = customer waiting time (in minutes) under the existing system

Y = customer waiting time (in minutes) under the new system

Then, X ~ N(µ1, ?12) and Y ~ N(µ2, ?22), where ?12 = ?22 = ?2, say and ?2 is unknown.

Claim:

The new system is superior to the existing system with respect to the mean waiting time.

Hypotheses:

Null: H0: µ1 = µ2 Vs Alternative: HA: µ1 > µ2

Test Statistic:

t = (Xbar - Ybar)/[s?{(1/n1) + (1/n2)}] where

s2 = {(n1 – 1)s12 + (n2 – 1)s22}/(n1 + n2 – 2);

Xbar and Ybar are sample averages and s1, s2 are sample standard deviations based on n1 observations on X and n2 observations on Y respectively.

Calculations

Summary of Excel calculations is given below:

Given

n1 =

1000

n2 =

500

Xbar =

4.2

Ybar =

3.5

s1 =

2.1

s2 =

1.9

s^2 =

4.143511

s =

2.035562

tcal =

6.27846

? =

0.05

tcrit =

1.645871

p-value =

2.23E-10

Distribution, Critical Value and p-value:

Under H0, t ~ tn1 + n2 - 2. Hence, for level of significance ?%, Critical Value = upper ?% point of tn1 + n2 - 2 and p-value = P(tn1 + n2 - 2 > tcal).

Using Excel Functions, the above are found to be as given in the above table.

Decision:

Since tcal > tcrit, H0 is rejected, which is also confirmed by the very low p-value, much less than the given level o f significance.

Conclusion:

There is sufficient evidence to suggest that the claim is valid.

=> The new system is superior to the existing syatem and hence the company may change over to the new system. ANSWER

DONE