Question

Customer Distribution by Weekday: A drop-in auto repair shop staffs the same number of mechanics on every weekday (weekends are not counted here). One of the mechanics thinks this is a bad idea because he suspects the number of customers is not evenly distributed across these days. For a sample of 289 customers, the counts by weekday are given in the table.

Number of Customers by Day (n = 289)

	Monday	Tuesday	Wednesday	Thursday	Friday
Count	45	44	59	75	66

The Test: Test the claim that the number of customers is not evenly distributed across the five weekdays. Test this claim at the 0.05 significance level.

(a) What is the null hypothesis for this test in terms of the probabilities of the outcomes?

H₀: p_mon = 0.45, p_tue = 0.44, p_wed = 0.59, p_thur = 0.75, p_fri = 0.66.

H₀: At least one of the probabilities doesn't equal 1/5.     

H₀: p_mon = p_tue = p_wed = p_thur = p_fri = 1/5.

H₀: None of the probabilities are equal to 1/5.

(b) What is the value of the test statistic? Round to 3 decimal places unless your software automatically rounds to 2 decimal places.

χ<sup>2=?

(c) Use software to get the P-value of the test statistic. Round to 4 decimal places unless your software automatically rounds to 3 decimal places.
P-value = ?

(d) What is the conclusion regarding the null hypothesis?</sup>

reject H₀

fail to reject H₀    

(e) Choose the appropriate concluding statement.

We have proven that the number of customers is evenly distributed across the five weekdays.

The data supports the claim that the number of customers is not evenly distributed across the five weekdays.    

There is not enough data to support the claim that the number of customers is not evenly distributed across the five weekdays.

Answer 1

H₀: p_mon = p_tue = p_wed = p_thur = p_fri = 1/5.

b)

applying chi square goodness of fit test:

	relative	observed	Expected	residual	Chi square
category	frequency(p)	Oi	Ei=total*p	R2i=(Oi-Ei)/√Ei	R2i=(Oi-Ei)2/Ei
1	0.200	45.000	57.80	-1.68	2.835
2	0.200	44.000	57.80	-1.82	3.295
3	0.200	59.000	57.80	0.16	0.025
4	0.200	75.000	57.80	2.26	5.118
5	0.200	66.000	57.80	1.08	1.163
total	1.000	289	289		12.4360
test statistic X2 =		12.436

c)

p value =

0.0144

d)

reject H₀

_e)The data supports the claim that the number of customers is not evenly distributed across the five weekdays.