A restaurant called Real Ramen recently opened in Ithaca, New York. The restaura
ID: 3151928 • Letter: A
Question
A restaurant called Real Ramen recently opened in Ithaca, New York. The restaurant offers several types of ramen and ramen-type noodle dishes, designated by the following numbers:
1 (Tokyo style ramen), 2 (Sapporo style ramen), 3 (Hakata style ramen), 4 (Abura soba), 5 (Maza-men), 6 (Tsuke-men), and 7 (Tantan-men).
The restaurant’s owners have asked noted Cornell University professor (and chef) Robert Johnson to offer advice regarding restaurant operations. The owners believe the dishes will be ordered in the following proportions: 1 (25%), 2 (20%), 3(10%), 4 (15%), 5 (5%), 6 (10%), and 7 (15%). The owners provided the number of orders placed for each dish during the first month of operations: 200 (1), 145 (2), 110 (3), 70 (4), 40 (5), 80 (6), and 105 (7).
Conduct the appropriate hypothesis test to determine whether the owners’ belief regarding the order proportions is correct. What is the p-value associated with the test statistic? (If you cannot calculate the p-value, place the best bounds upon the p-value that you can find.)
Explanation / Answer
Given that n1 = 7 n2 = 7
X
(x-X)²
y
(y-Y)²
0.25
0.0115
200
8622.4410
0.20
0.0033
145
1433.1600
0.10
0.0018
110
8.1630
0.15
0.0001
70
1379.5950
0.05
0.0086
40
4508.1690
0.10
0.0018
80
736.7370
0.15
0.0001
105
4.5920
Total 1
0.0272
750
16692.8560
X = 0.1429 Y = 107.1429
S2 = 1/n1+n2-2( (x-X)²+ (y-Y)²) =1/12 (0.0272 + 16692.8560)=1391.0736
The null hypothesis is given by
H0 : µx = µyi.e., to determine the owner’s belief regarding the order proportions is correct
Against the alternative hypothesis
H1 : µx µy i.e., to determine the owner’s belief regarding the order proportions is not correct
The test statistic is given by
t = X - Y/s2/(1/n1)+(1/n2) tn1+n2-2
t = 0.1429 - 107.1429/(1391.0736)/(1/7)+(1/7) t12
t = 107/69.7713
tcal = 1.5336
the tabulated at 0.05 level of significance for two tailed test is 2.18 i.e., ttab = 2.18
here tcal > ttab so we reject the null hypothesis at 5% level of significance
therefore we conclude that to determine the owner’s belief regarding the order proportions is correct. P value is 0.15106 the result is not significant at p<0.05
X
(x-X)²
y
(y-Y)²
0.25
0.0115
200
8622.4410
0.20
0.0033
145
1433.1600
0.10
0.0018
110
8.1630
0.15
0.0001
70
1379.5950
0.05
0.0086
40
4508.1690
0.10
0.0018
80
736.7370
0.15
0.0001
105
4.5920
Total 1
0.0272
750
16692.8560
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