Question

A test is given in to a group of 20 subjects, with X= 105.5 and S=5.8, and then the test is administered to another group of 32 subjects with X= 107.2 and S=5.2. The true population variances are unknown but assumed to be equal. You wish to test whether the means are equal or not (i.e., a two-sided test). At a=0.05, conduct the test using p-value and state your conclusions.

Answer 1

Null hypothesis : There is no significant difference between the population means of both the groups.

i.e, the population mean of first group = population mean of second group .

Alternative hypothesis : There is significant difference between the population means of both the groups.

Assuming variance to be equal,

Test statistic is given by -

Where, is the sample mean of first group = 105.5

is the sample mean of second group = 107.2

S1 is the sample standard deviation of the first group = 5.8

S2 is the sample standard deviation of the second group = 5.2

n1 = sample size of the first group = 20

n2 = sample size of the second group = 20

sp is the Pooled standard deviation given by -

  

= 5.51

Hence, the value of the test statistic will be -

  

= -0.98

P value = P (|t| > 0.98) = Area under the t curve with 38 degrees of freedom on the left side of t = -0.98 and right side of t = 0.98

It can be obtained using the formula =TDIST(0.98,38,2) in excel.

So, P value = 0.333

Since P value (0.333) > level of significance (0.05), we may not reject the null hypothesis.

Hence, there is significant difference between the population means of both the groups, that is, the population means of both the groups are equal.