Construct a 95% confidence interval for μ1 - μ2. Two samples are randomly selected from normal populations. The sample statistics are given below. n1 = 8n2 = 71 = 4.12 = 5.5 s1 = 0.76s2 = 2.51
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n1 = 8, n2 = 71
/x1 = 4.12, /x2 = 5.5
s1 = 0.76, s2 = 2.51
s1/s2 = 0.76 / 2.51 = 0.302788844 < 1/2, so we can assume unequal variances.
We will need to make a small sample inference, since n1 is small.
v
= [((s1)^2 / n1) + ((s2)^2 / n2)]^2 / [(1 / (n1 - 1))*((s1)^2 / n1)^2 + (1 / (n2 - 1))*((s2)^2 / n2)^2]
= [((0.76)^2 / 8) + ((2.51)^2 / 71)]^2 / [(1 / (8 - 1))*((0.76)^2 / 8)^2 + (1 / (71 - 1))*((2.51)^2 / 71)^2]
= 0.025899688 / [0.000744691 + 0.000112481]
= 0.025899688 / 0.000857172
= 30.2152664
~ 30
estimated st dev
= sqrt[((s1)^2 / n1) + ((s2)^2 / n2)]
= sqrt[((0.76)^2 / 8) + ((2.51)^2 / 71)]
= 0.401165555
95% confidence interval for miu1 - miu2
= (/x1 - /x2) +/- t(v , 0.025) * estimated st dev
= (4.12 - 5.5) +/- t(30, 0.025) * (0.401165555)
= -1.38 +/- (2.042)*(0.401165555)
= -1.38 +/- 0.819180063
= [-2.199180063 , -0.560819937]
/x1 = 4.12, /x2 = 5.5
s1 = 0.76, s2 = 2.51
s1/s2 = 0.76 / 2.51 = 0.302788844 < 1/2, so we can assume unequal variances.
We will need to make a small sample inference, since n1 is small.
v
= [((s1)^2 / n1) + ((s2)^2 / n2)]^2 / [(1 / (n1 - 1))*((s1)^2 / n1)^2 + (1 / (n2 - 1))*((s2)^2 / n2)^2]
= [((0.76)^2 / 8) + ((2.51)^2 / 71)]^2 / [(1 / (8 - 1))*((0.76)^2 / 8)^2 + (1 / (71 - 1))*((2.51)^2 / 71)^2]
= 0.025899688 / [0.000744691 + 0.000112481]
= 0.025899688 / 0.000857172
= 30.2152664
~ 30
estimated st dev
= sqrt[((s1)^2 / n1) + ((s2)^2 / n2)]
= sqrt[((0.76)^2 / 8) + ((2.51)^2 / 71)]
= 0.401165555
95% confidence interval for miu1 - miu2
= (/x1 - /x2) +/- t(v , 0.025) * estimated st dev
= (4.12 - 5.5) +/- t(30, 0.025) * (0.401165555)
= -1.38 +/- (2.042)*(0.401165555)
= -1.38 +/- 0.819180063
= [-2.199180063 , -0.560819937]