Grocery Store Statistics

Date: 06/10/98 at 21:39:03
From: David Delight
Subject: Grocery store statistics

The manager of a grocery store chain is concerned about the quality of 
service, as measured by service times. A sample of service times from 
two stores is given below.

         Store 1                      Store 2
   x1 (barred) = 12 min         x2(barred) = 10 min 
   S1^2 = 1.9 min               S2^2 = 2.5 min
   N1 = 7                       N2 = 10   
  
1. Develop a 90% confidence interval on the population variance for 
   the service times for store 1.

2. Develop a 95% confidence interval on the population for the service 
   times for store 2.

3. Test to determine if there is a difference between the service time 
   variances for the two stores.

4. Test to determine if the variance times for store 2 is greater than 
   the variance times for store 1.

Date: 06/11/98 at 12:38:25
From: Doctor Anthony
Subject: Re: Grocery store statistics

There is some doubt about the information you provide. You quote S1^2 
as 1.9 min, but if you mean variance then the units are min^2. I shall 
be assuming you do mean the variance is 1.9, but if it should be the 
standard deviation, then you will have to rework the problem making 
the necessary adjustments. I shall be using X^2 to mean chi-squared, 
and s^2 to be population variance.

Problem 1:

To find a 90% confidence interval for population variance use the 
chi-squared distribution as follows:

We use the inequality:

   (N1-1)*S1^2         (N1-1)*S1^2
   ----------- < s^2 < -----------
     X^2(5%)             X^2(95%)

In this case the degrees of freedom for X^2 is equal to (N1-1). Then:

   6 (1.9)         6 (1.9)
   ------- < s^2 < -------
    12.59           1.64

   .9055   < s^2 < 6.95  

Problem 2:

Here, degrees of freedom = 10 - 1 = 9.

   (N2-1)S2^2          (N2-1)S2^2
   ----------  < s^2 < ----------
    X^2(2.5%)          X^2(97.5%)

   9 (2.5)        9 (2.5)
   ------- < s^2 < -------
    19.02           2.7

   1.183   <  s^2  <  8.333

Problem 3:

This problem is similar to question (4) below, except that we do a 
two-tailed test rather than a one-tailed test. For this we use F 
ratio test:

        larger s^2    2.5
   F =  ----------- = --- = 1.316
        smaller s^2   1.9

The degrees of freedom are 9 and 6 so we test our value 1.316 against 
the tabular value F(2.5%)(9,6) = 4.83.

Our value of 1.316 is therefore not significant and there is no 
evidence of a difference in variance at the 5% level of significance.

[Note that we used F(2.5%) here instead of F(5%) as in question (4).  
In other words we have 2.5% in each tail giving a total 5% tail area.]

Problem 4:

Again, for this we use F ratio test.

        larger s^2    2.5
   F =  ----------- = --- = 1.316
        smaller s^2   1.9

The degrees of freedom are 9 and 6 so we test our value 1.316 against 
the tabular value F(5%)(9,6) = 4.10.

The value 1.316 is clearly not significant and we conclude that the 
variance of store 2 is not necessarily greater than store 1.

-Doctor Anthony,  The Math Forum
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