A product is packaged on two separate production lines. Let X and Y be the fill
ID: 3135449 • Letter: A
Question
A product is packaged on two separate production lines. Let X and Y be the fill weight in grams when a package is filled on respectively the first line or the second line. Assume that the distributions of X and Y are normal with an unknown variance that is the same on both lines. A dozen packages were sampled from each line and weighed, giving the following results:
From the first line: 1071 1076 1070 1083 1082 1067 1078 1080 1075 1084 1075 1080
From the second line:1074 1069 1075 1067 1068 1079 1082 1064 1070 1073 1072 1075
Explanation / Answer
Given that n1 = 12 n2 = 12
X
(x-X)²
y
(y-Y)²
1071
1074
33.0625
2.7889
1076
1069
0.5625
11.0889
1070
1075
45.5625
7.1289
1083
1067
39.0625
28.4089
1082
1068
27.5625
18.7489
1067
1079
95.0625
44.4889
1078
1082
1.5625
93.5089
1080
1064
10.5625
69.3889
1075
1070
3.0625
5.4289
1084
1073
52.5625
0.4489
1075
1072
3.0625
0.1089
1080
1075
10.5625
7.1289
12921
12868
322.2500
288.6668
X = 1076.75 Y = 1072.33
S2 = 1/n1+n2-2( (x-X)²+ (y-Y)²) =1/22(322.2500+288.6668)=27.7689
The null hypothesis is given by
H0 : µx = µyi.e., to claim that the mean weights form the lines are equal
Against the alternative hypothesis
H1 : µx µy i.e., to claim that the mean weights form the lines are not equal
The test statistic is given by
t = X - Y/s2/(1/n1)+(1/n2) tn1+n2-2
t = 1076.75- 1072.33/(27.7689)/(1/12)+(1/12) t22
t = 4.42/2.1509
tcal = 2.0550
the tabulated t0.10 for (22)d.f for two tailed test is 1.72 i.e., ttab = 1.72
here tcal > ttab so we reject the null hypothesis at 0.01 level of significance
therefore we conclude that the mean weight form the lines are not equal
the p value is given by 0.0519
X
(x-X)²
y
(y-Y)²
1071
1074
33.0625
2.7889
1076
1069
0.5625
11.0889
1070
1075
45.5625
7.1289
1083
1067
39.0625
28.4089
1082
1068
27.5625
18.7489
1067
1079
95.0625
44.4889
1078
1082
1.5625
93.5089
1080
1064
10.5625
69.3889
1075
1070
3.0625
5.4289
1084
1073
52.5625
0.4489
1075
1072
3.0625
0.1089
1080
1075
10.5625
7.1289
12921
12868
322.2500
288.6668
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