In: Statistics and Probability
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]
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.