Question

I have Standard Deviation and Mean of 2 sets of data.
Based on the data, how can we infer at the 5% significance level that the score of individuals in the 4th year is better than the individuals in 1st year?

average	71.29	76.98
S.D.	8.58	8.119
	Year 1	Year 4

The sample size is 430

Answer 1

Answer: In order to determine if the score of the individuals in 4th year is better than the score of the individuals in 1st year, we construct our null and alternative hypotheses as H0: mu1 = mu2 vs Ha: mu1 < mu2 where mu1 and mu2 are the unknown means of the scores of individuals of 1st year and 4th year respectively.

The test statistic used for this test is T= (x1bar-x2bar)/sqrt((s1*s1/n1)+(s2*s2/n2)) where x1bar, x2bar are the sample means, s1,s2 are the sample standard deviations and n1,n2 are the sample sizes. sqrt refers to the square root function.

We reject H0 if T(observed) < -t(alpha,v) where t(alpha,v) is the upper alpha point of a Student's t distribution with "v" degrees of freedom. Alpha is the level of significance. v = (((s1/n1)+(s2/n2))^2) / (((s1/n1)^2)/(n1-1))+(((s2/n2)^2)/(n2-1))

Here T(observed) = -9.988622 and -t(alpha,v) =  -1.646637. Thus we see that T(observed) < - t(alpha,v). Thus we reject H0 and conclude on the basis of the given sample measures at a 5% level of significance that the scores of the individuals in 4th year is better than the score of the individuals in 1st year

[The answers are obtained using probability tables of Student's t distribution]