Size (Thousands of Square Feet)

252soln7a (11/1/99)

PROBLEM E.1. (Sincich}An Ernst and Young survey of 126 warehouses operated by retail stores tests the independence of the number of deliveries to stores per week to warehouse size. Use for a test of independence.

Size (thousands of square feet)

Deliveries/week Below 100 100-249.9 250-400Above 400

1 or fewer 513 9 5

2-3121113 6

4-5 9141311

Solution: This is an extremely basic chi-squared problem with . First we total rows and columns and then compute the fraction of data in each row. For example in the first row there are 32 deliveries out of a total of 126, so the fraction in the first row is 126/30=.3540, which is the first element in . (Observed) is the array in the frame.

We use to get (Expected) by multiplying the column totals. For example we get 6.6032 by multiplying the first column total, 26, by .2540. Row and column totals remain the same except for rounding error.

We now place corresponding values of and together to get . Degrees of freedom are , where is the number of rows and is the number of columns.

. We compare this with from the table. Since our computed is less than the table , we cannot reject .

252soln7b (13/17/00)

PROBLEM E.2. A random sample of 64 cans of each of 3 brands of canned fruit is examined. The proportion that are not as labeled is .1094 for brand 1, .0781 for brand 2 and .1563 for brand 3. Is the proportion the same for each brand?

Solution:We are testing or , where is the proportion not as labeled in batch1, etc

Since the quantities in must be whole numbers, we get the first row of by multiplying 64, the batch size, by .1094 for brand 1, .0781 for brand 2 and .1563 for brand 3 and taking the nearest integer. The first value is thus and we use 7. Our table is thus the table at right.

We use our row proportions to create at right.

We now place corresponding values of and together to get Degrees of freedom are , where is the number of rows and is the number of columns. . We compare this with from the table. Since our computed is less than the table , we cannot reject .

PROBLEM E.3. A real estate firm wants to check whether selling price is related to the number of days a home is on the market. A random sample of 100 homes is taken and divided into three classes according to selling price. The realtor discovers that 57% of the 30 homes in the under $100,000 class were on the market for 60 days or fewer. 38% of the 50 homes in the $100,000 - $200,000 class were on the market for 60 days or fewer. Finally, in the above $200,000 class, 35% of 20 homes were on the market for 60 days or fewer.

a. Do a test of the equality of proportions for the $100,000-$200,000 class and the above $200,000 class. Repeat this test as a chi-squared test.

1. Do a test of equality of proportions for all three classes.

Solution: Our data is and .

a) Let and Then our hypotheses are and .

252soln7c (11/8/99)

. Since this is between we cannot reject .

Note:To do a test, use the numbers in columns 2 and 3 below, but instead of using , use .

b) or

Solution:We are testing or , where is the proportion sold in 60 days among the expensive homes, is the same proportion among midrange homes, etc.

To get the top line of , multiply and by and respectively .As in the previous problem the results must be whole numbers.

We use our row proportions to create at right.

We now place corresponding values of and together to get Degrees of freedom are , where is the number of rows and is the number of columns. . We compare this with from the table. Since our computed is less than the table , we cannot reject .