Grocery Store StatisticsDate: 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 Check out our web site! http://mathforum.org/dr.math/ |
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