Question

Trends in Restaurant Take-Out

A local restaurant does a pretty significant take-out business. According to the first few years that the business was open, the following table gives the percentages of take-out orders happen each day of the week.

Sunday	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday
14%	9%	8%	11%	15%	21%	22%

They recently noticed that they have been short drivers some days but have been not as busy other days. In a pool of 600 orders, this is the distribution of the number of to-go orders that occurred on each day.

Sunday	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday
120	80	45	82	60	80	133

Test the hypothesis that the original distribution is followed at the 5% level of significance. Does the test suggest that the distribution based on recent data has changed? Show all calculations and be sure to list all important elements of the hypothesis test (Null and alternative hypotheses, critical region, test statistic, decision, and conclusion).

Answer 1

Category	Observed Frequency (O)	Expected Frequency (E)	(O-E)²/E
Sunday	120	600 * 0.14 = 84	(120 - 84)²/84 = 15.4286
Monday	80	600 * 0.09 = 54	(80 - 54)²/54 = 12.5185
Tuesday	45	600 * 0.08 = 48	(45 - 48)²/48 = 0.1875
Wednesday	82	600 * 0.11 = 66	(82 - 66)²/66 = 3.8788
Thursday	60	600 * 0.15 = 90	(60 - 90)²/90 = 10
Friday	80	600 * 0.21 = 126	(80 - 126)²/126 = 16.7937
Saturday	133	600 * 0.22 = 132	(133 - 132)²/132 = 0.0076
Total	600	600	58.8146

Null and Alternative hypothesis:                  
Ho: Proportions are same.                  
H1: Proportions are different.                  

Test statistic:                  
χ² = ∑ ((fo-fe)²/fe) = 58.8146              

df = n-1 = 6              

Critical value:                  
χ²α = CHISQ.INV.RT(0.05, 6) = 12.5916   

Reject ho, if χ² > 12.5916.

Decision:                  
χ² =58.8146 > 12.5916, Reject the null hypothesis                  

Conclusion:

There is enough evidence to conclude that the distribution based on recent data has changed at 0.05 significance level.