Let X_1, X_2,..., X_m tilde Distribution_1(mu x, sigma^2_x) and Y_1, Y_2, ..., Y
ID: 3152433 • Letter: L
Question
Let X_1, X_2,..., X_m tilde Distribution_1(mu x, sigma^2_x) and Y_1, Y_2, ..., Y_n tilde Distribution_2(mu gamma, sigma^2_gamma) be two independent random samples from two distributions In usual notation. X bar = 4, s^2_x = 9, Y bar = 12 and s^2_gamma = 4. Perform hypotheses testing at significance level of alpha = 0.05, for each case below. Case 1: m = 9, n = 16, sigma^2_x notequalto sigma^2_gamma and two distributions are normal. Case 2: m = 16, n = 25, sigma^2_x = sigma^2_gamma and two distributions are normal. Case 3: m = 81, n = 49, sigma^2_x notequalto sigma^2_gamma and two unknown distributions. Case 4: m = 34, n = 44, sigma^2_x = sigma^2_gamma and two unknown distributions.Explanation / Answer
X-=4, S(x)^2= 9, Y_=12 and S(y)^2=4
1) Ans m=9, n=16, Smaple are getting from two different variance.
The sample are generated from two different variance with small sample sizes. Therefore the test statistic is
t=(x_-y_)/sqrt[{S(x)^2}/m+{S(y)^2}/n] =(4-12)/sqrt(9/9+4/16)=-8/1.118=-7.155
P(t>7.155)+P(t<7.155)=0.0000
The corresponding p-value for t with df=(9+16-2)=23 is 0.0000
2. Ans;
The sample are generated from same variance with small sample sizes.
S^2=[S(x)^2*(m-1)+S(y)^2*(n-1)]/(m+n)=(9*15+4*24)/39=5.923
Therefore the test statistic is
t=(x_-y_)/[S*sqrt(1/m+1/n)] =4-12/(2.4337*sqrt(1/16+1/25)=-8/0.7792=-10.2669
The P(t>-10.2669)=1
The corresponding p-value for t with df=(16+25-2)=39 is 1.000
3) Ans m=81, n=45, Smaple are getting from two different variance.
The sample are generated from two different variance with enough sample sizes for Normal test. Therefore the test statistic is
Z=(x_-y_)/sqrt[{S(x)^2}/m+{S(y)^2}/n] =-8/sqrt[9/81+4/45]=-8/0.447=-17.897
P(Z<-17.897)+P(Z>17.897)=0.0000
The corresponding p-value for t with df=(81+45-2)=124 is 0.0000
4. Ans;
The sample are generated from same variance with small sample sizes.
S^2=[S(x)^2*(m-1)+S(y)^2*(n-1)]/(m+n)=(9*33+4*43)/(34+44-2)=6.17
Therefore the test statistic is
Z=(x_-y_)/[S*sqrt(1/m+1/n)] =4-12/(2.4842*sqrt(1/34+1/44)= -5.7216
P(Z>-5.7216)=1
The corresponding df=(134+44-2)=76
Hypothesis S^2 df Quantile Test Statistic p-value Conclusion mu(X)-mu(Y)=0 23 t(0.025)=-2.33 -7.155 0.000 Reject Ho mu(X)-mu(Y)=/0Related Questions
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