Answer to Question #250324 in Statistics and Probability for smilynne

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 Public

_ _

X1 = 16,400 X2 = 15,170

S1 = 700 S1 = 800

n1 = 15 n2 = 9

Answer 1

$n_1= 15 \\ \bar{x_1} = 16400 \\ s_1 = 700 \\ n_2 = 9 \\ \bar{x_2} = 15170 \\ s_2 = 800 \\ H_0: \mu_1=\mu_2 \\ H_1: \mu_1 ≠ \mu_2$

Test-statistic:

$t = \frac{\bar{x_1} - \bar{x_2}}{\sqrt{s^2_{p} \times (\frac{1}{n_1} + \frac{1}{n_2})}} \\ s^2_{p} = \frac{(n_1-1) \times s^2_1 + (n_2-1) \times s^2_2}{n_1+n_2-2} \\ s^2_p = \frac{14(700)^2 + 8(800)^2}{15+9-2} \\ = \frac{6860000 + 5120000}{22} \\ = 544545.45 \\ t = \frac{16400 -15170}{\sqrt{544545.45 \times (\frac{1}{15} + \frac{1}{9})}} \\ = \frac{1230}{311.139} \\ = 3.953 \\ df= n_1+n_2-2 = 15+9-2=22 \\ α=0.01$

Tabulated value

$t_{22,0.005} = 2.819$

Reject H0 if |t| ≥ 2.819

Conclusion: $t=3.953>t_{tab} = 2.819$

We reject H₀ at 0.01 level of significance.

There is sufficient evidence that the private school teachers does not receive the same salary with the public school teachers.