A sample of 10 twins was taken In each case one twin started

A sample of 10 twins was taken. In each case, one twin started at a community college before finishing their education at ETSU; the other twin went directly to ETSU. The difference in their CCTST, a critical thinking exit exam that all ETSU students are required to take, averaged 12 points with a SD of 5. (Students who started at a community college scored higher on average). Test the null hypothesis that there is no difference in the performances against the alternative that performance is worse for students who go directly to ETSU. Use alpha = 0.01. Make sure to identify the population group.

Solution

here a sample of 10 twins are taken.one twin started at a community college before finishing their education at ETSU ; the other twin went directly to ETSU

let X denote the random variable denoting the points in CCTST for the twin who started community college and Y denote the same for the twin went directly at ETSU

since here the sample is of twins there must be some relation between X and Y. so here the population is

X and Y follows a bivariate normal distribution with parameters u1=E[X], u2=E[Y], s1²=Var(X),s2²=Var(Y), r=correlation coefficient

here null hypothesis is there is no difference in the performance. so H0: u1-u2=0

alternative is the performance is worse for students who go directly to ETSU. so u1>u2 or u1-u2>0

so H1:u1-u2>0

we have a sample of size n=10.

let Xbar be the sample mean of X and Ybar be that of Y. and let s be the sample SD of Xbar-Ybar

given that Xbar-Ybar=12 and s=5

then the test statistic is T=(xbar-ybar)*sqrt(n)/s which under H0 follows a t distribution with df n-1=9

we reject H0 iff t>t_alpha;9   where t_alpha;9 is the upper alpha point of a t distribution with df 9 and t is the observed value of T and alpha is the level fo significance

given that alpha=0.01 so t_0.01;9=2.82144   t=12*sqrt(10)/5=7.589

hence t>t_0.01;9

hence H0 is rejected. hence the conclusion is that at 1% level of signifiance based on the data at hand the performance is worse for students who go directly to ETSU