Answer to Question #250326 in Statistics and Probability for Goldie

Question

A researcher wishes to determine whether the monthly salary of professional elementary teachers in private schools and elementary teachers in public schools differs. He selects a sample of elementary teachers. From each type of school calculate the means and standard deviations of their salaries. At 0.01 alpha level, can he conclude that the private school teachers does not receive the same salary with the public school teachers? Assume that the populations are approximately normally distributed. 

Private

_

X1 = 16,400

S1 = 700

n1 = 15

Public

_

X2 = 15,170

S1 = 800

n2 = 9

Answer 1

Solution:

Private-

$X_1 = 16,400 \\S_1 = 700 \\n_1 = 15$

Public-

$X_2 = 15,170 \\S_2 = 800 \\n_2 = 9$

$H_0:\mu_1=\mu_2 \\H_1:\mu_1\ne \mu_2$

Poole standard deviation:

$S=\sqrt{\dfrac{(n_1-1)S_1^2+(n_2-1)S_2^2}{n_1+n_2-2}} \\=\sqrt{\dfrac{(15-1)700^2+(9-1)800^2}{15+9-2}}$

$=737.933$

Test statistic$=t=\dfrac{\bar X_1-\bar X_2}{S\sqrt{\dfrac1{n_1}+\dfrac1{n_2}}}$

$=\dfrac{16400-15170}{737.933\sqrt{\dfrac1{15}+\dfrac1{9}}} \\=3.95$

t-value at df = 15+9-2 = 23 at 0.01 level is 2.5

Our test statistic is greater than 2.5.

So, we reject null hypotheses.

Thus, $\mu_1\ne \mu_2$ and he can conclude that the private school teachers does not receive the same salary with the public school teachers.