Question

In: Statistics and Probability

A local franchise of a national chain of day-old pastry stores recorded the number of customers...

A local franchise of a national chain of day-old pastry stores recorded the number of customers that came in to the store for five weeks this fall. These data are recorded in the attached file. The national franchisor states that the percentage of customers is distributed as follows:   Sunday – 20%, Monday – 8%, Tuesday – 6%, Wednesday – 7%, Thursday – 12%, Friday – 22%, and Saturday – 25%. The manager doesn’t trust the franchisor and wants to show his store has different percentages. Use a χ² Goodness-of-Fit test to determine if the manager has evidence that his store is varies from the national percentages.

Customers per day
First five weeks after September
Sun 53
Mon 22
Tue 18
Wed 24
Thu 36
Fri 51
Sat 83
Sun 69
Mon 14
Tue 19
Wed 24
Thu 37
Fri 57
Sat 68
Sun 60
Mon 23
Tue 16
Wed 22
Thu 34
Fri 60
Sat 72
Sun 68
Mon 22
Tue 17
Wed 25
Thu 39
Fri 64
Sat 83
Sun 84
Mon 25
Tue 15
Wed 19
Thu 36
Fri 60
Sat 81
1500

Solutions

Expert Solution

SOLUTION

The Hypothesis:

H0: The sample has a distribution that agrees with the national percentage.

Ha: The sample has a distribution that is different from the national percentage.

The Test Statistic:

Each Expected value = (% / 100) * 1500

Observed Expected % Expected (O-E)2 (O-E)2/E
Sunday 334 20 300 1156 3.85333
Monday 106 8 120 196 1.63333
Tuesday 85 6 90 25 0.27778
Wednesday 114 7 105 81 0.77143
Thursday 182 12 180 4 0.02222
Friday 292 22 330 1444 4.37576
Saturday 387 25 375 144 0.38400
Total 1500.00 100.00 1500.00 11.318

test = 11.318

The Critical Value: The critical value at = 0.05, df= n - 1 = 6

critical = 11.07

The p value: The p value: The p value at test = 11.318, df = 6; P value = 0.0454

The Decision Rule:

The Critical Value Method: If test is > critical, then Reject H0.

The p - value Method: If p value is < , Then Reject H0.

The Decision:  

The Critical Value Method: Since test (11.318) is > critical (11.07), We Reject H0.

The p - value Method: Since p value (0.0454) is < (0.05), We Reject H0.

The Conclusion: There is sufficient evidence at the 95% significance level to conclude that sample has a distribution that is different from the national percentage.

orchestra answered 2 years ago
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