Question

Restaurants usually have more customers closer to Saturday than earlier in the week. A restaurant chain hypothesizes that, of all the customers in a week, only 7% come in on each of the 4 days, Sunday through Wednesday, whereas 21% come in on Thursday, 25% on Friday, and 30% on Saturday.

The restaurant collects data over a random sample of days and obtains the following customer count. Management want to test the hypothesis that the distribution of customers over the days of the week is as they expect. Sunday 100, Monday 130, Tuesday 125, Wednesday 130, thursday 310, friday 370, saturday 410.

(a) Construct the table of expected frequencies.

(b) Write the null and alternative hypotheses. State the requirements and verify whether they are met.

(c) Suppose we reject the null hypothesis. Write the verbal conclusion in context.

Answer 1

Chi square test for Goodness of fit  
total frequency=   1575
expected frequncy,E = expected proportions*total frequency  

a)

category	observed frequencey, O	expected proportion	expected frequency,E
Sunday	100	0.0700	110.250
Monday	130	0.0700	110.250
Tuesday	125	0.0700	110.250
Wednesday	130	0.0700	110.250
Thursday	310	0.2100	330.750
Friday	370	0.2500	393.750
Saturday	410	0.300	472.500

b)

Ho: distribution of customers over the days of the week is as they expect

H1: distribution of customers over the days of the week is not as they expect

requirement: expected frequency should be at least five

requirement is met.

c)

chi square test statistic,X² = Σ(O-E)²/E =   21.0038              
                  
level of significance, α=   0.05              
Degree of freedom=k-1=   7   -   1   =   6
                  
P value =   0.0018   [ excel function: =chisq.dist.rt(test-stat,df) ]          
Decision: P value < α, Reject Ho                  

conclusion: there is enough evidence to conclude that distribution of customers over the days of the week is not as they expect