Question

In: Statistics and Probability

5.5. A large supermarket carries four qualities of beef. Customers are believed to purchase these four...

5.5. A large supermarket carries four qualities of beef. Customers are believed to purchase these four qualities with probabilities of 0.10, 0.30, 0.35, and 0.25, respectively, from the least to most expensive. A sample of 500 purchases resulted in sales of 48, 162, 191, and 99 of respective qualities. Does this sample contradict the expected proportions? Use a) α = 0.05 b) α =0.025 [5+5 = 10]

Solutions

Expert Solution

Step 1:

Ho: The proportions of different qualities of beef purchased are : p1= 0.10, p2 = 0.30, p3= 0.35, and p4=0.25

Ha: Some of the population proportions differ from the values stated in the null hypothesis

This corresponds to a Chi-Square test for Goodness of Fit.

Step 2: test statistics

Subjects Observed values
(fo)
Expected Proportions Expected values
(fe)
(fo-fe)2/ fe
type 1 48 0.100 50.000 0.080
type 2 162 0.300 150.000 0.960
type 3 191 0.350 175.000 1.463
type 4 99 0.250 125.000 5.408
Total 500 1.000 500.000 7.911

= 7.911

Step 3:

(a) = 0.05

df = 3, critical = Chi square critical = CHISQ.INV.RT(probability,df) = CHINS.INV.RT(0.05, 3)= 7.815

As (7.911) is greater than critical, we reject the Null hypothesis.

Hence the sample contradicts the expected proportions.

(b) = 0.025

df = 3, critical = Chi square critical = CHISQ.INV.RT(probability,df) = CHINS.INV.RT(0.025, 3)= 9.348

As (7.911) is less than critical, we fail to reject the Null hypothesis.

Hence we do not have sufficient evidence to believe that the sample proportions are different that the proportions states in Ho.


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