Question

source: m. j. tramo, w. c. lodtus, t. a. stukel, r. l. green, j. b. weaver, and m. s. gazzaniga. “brain size, head size, and intelligence quotient in monozygotic twins.” neurology 50, 1246 - 52.
let $mu_1$ and $mu_2$ represent the mean full scale iq score for all first - born identical twins and second - born identical twins, respectively, and let $mu_d$ be the mean of the differences in iq score of all identical twins (iq score of first - born twin minus iq score of second - born twin).
which of the following are the appropriate null and alternative hypotheses?
a. $mu_0:mu_d = 0$ $mu_a:mu_d>0$
b. $mu_0:mu_d = 0$ $mu_a:mu_d<0$
c. $mu_0:mu_d>0$ $mu_a:mu_d = 0$
d. $mu_0:mu_1=mu_2$ $mu_a:mu_1>mu_2$

Explanation:

In hypothesis testing for paired data (here, differences in IQ scores of twins), the null hypothesis \(H_0\) typically assumes no difference (\(\mu_d = 0\)). The alternative hypothesis \(H_a\) depends on the research question. If we are interested in whether the first - born has a higher IQ (since \(d=\text{First - born IQ}-\text{Second - born IQ}\)), then \(H_a:\mu_d>0\).

Answer:

A. \(H_0:\mu_d = 0\), \(H_a:\mu_d>0\)