Question

Your professor wants to determine if her students today are performing better in statistics than 10 years ago. End of semester grades from a random sample of students enrolled today and one of students enrolled 10 years ago have been obtained:

Today 10 Years Ago
82 88
σ 2 112.5 54
n 45 36
The test statistic for the difference between the two population means is _____.
a) –3
b) –1.5
c) –.65
d) –.47

Answer 1

z = -3

Explanation:

If we had two independent samples, the variance is known and
`n_1` and
`n_2` are bigger than 30, the test statistic for the difference between the two population means is calculated as:

`z=\frac{x_1-x_2}{\sqrt{(s_1^2)/(n_1)+(s_2^2)/(n_2) } }`

Where
`x_1` and
`x_2` are the means of the samples,
`s_1^2` and
`s_2^2` are the variances of the samples and
`n_1` and
`n_2` are the size of the sample.

Finally, replacing
`x_1` by 82,
`x_2` by 88,
`s_1^2` by 112.5 ,
`s_2^2` by 54,
`n_1` by 45 and
`n_2` by 36, we get that the statistic is equal to:

`z=\frac{82-88}{\sqrt{(112.5)/(45)+(54)/(36) } }=-3`