Answer to Question #171760 in Statistics and Probability for Moon Shah

Question

Men’s heights are normally distributed with mean 180 cm and standard deviation 4 cm, while women’s heights are normally distributed with mean 175 cm and standard deviation 3 cm. In a sample of 100 married couples, the average difference between husband’s height and wife’s height is 3.4 cm. By performing an appropriate test, comment on if the choice of partner in marriage is influenced by considerations of height, at 1% level of significance. Show full working details for your test.

(15 marks)

Answer 1

$H_0: m_1-m_2=5\\ H_1: m_1-m_2<5\\ m_1\text{ --- average men's height in the population of married couples}\\ m_2\text{ --- average women's height in the population of married couples}\\ n=m=N=100\\ \sigma_1=4\text{ --- standard deviation of wives' heights}\\ \sigma_2=3\text{ --- standard deviation of husbands' heights}\\ \alpha=0.01\\ \overline{x}-\overline{y}=3.4\\ Z=\frac{\overline{X}-\overline{Y}}{\sqrt{\frac{\sigma_x^2}{n}+\frac{\sigma_y^2}{m}}}\\ z=\frac{3.4}{\sqrt{\frac{4^2}{100}+\frac{3^2}{100}}}=6.8\\ \Phi(z_{cr})=\frac{1-\alpha}{2}\\ \Phi(z_{cr})=0.495\\ z_{cr}=2.58\\ (-\infty,-2.58)\text{ --- rejection region}\\ z=6.8\text{ does not fall into the rejection region. So we accept }\\ H_0:m_1-m_2=5\\ \text{The choice of partner in marriage is not influenced by considerations of height}.$