Question

A particular female student has a mean height of 155 centimeters and a standard deviation of 20 centimeters, whilst a particular male student has a mean height of 170 centimeters and a standard deviation of 15 centimeters. What is the probability that a random sample of 36 female students will have a mean height lower than the mean height of 16 male students by at least 12.5 centimeters? 

Answer 1

x= female heights

y=male heights

$\mu_x=155 \\ \sigma_x=20 \\ n_1=36 \\ \mu_x=\mu_{\bar{x}}= 155 \\ \sigma_{\bar{x}}= \frac{\sigma_x}{\sqrt{n_1}} = \frac{20}{\sqrt{36}} = 3.33 \\ \mu_y=170 \\ \sigma_y=15 \\ n_2=16 \\ \mu_y=\mu_{\bar{y}}= 170 \\ \sigma_{\bar{y}}= \frac{\sigma_y}{\sqrt{n_2}} = \frac{15}{\sqrt{16}} = 3.75 \\ T= \bar{y} - \bar{x} \\ \mu_T=170-155 = 15 \\ \sigma_T^2 = (3.75)^2 + (3.33)^2 = 25.176 \\ \sigma_T = 5.01 \\ P(T>12.5) = 1 -P(T<12.5) \\ = 1 -P(Z< \frac{12.5-15}{5.01}) \\ = 1 -P(Z< -0.499) \\ = 1- 0.3088 \\ = 0.6912$