Question

some research suggests that first born children may have higher iq scores than their later born siblings. do first - born identical twins have higher iq scores than their second - born sibling? data from a 1998 study were analyzed to determine whether first - born identical twins have higher iq scores than their second - born siblings. ten pairs of adult identical twins were assessed and their full scale iq scores were calculated.
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. $\bigcirc$ $h_0:mu_d = 0$ $h_a:mu_d>0$
b. $\bigcirc$ $h_0:mu_d = 0$ $h_a:mu_d<0$
c. $\bigcirc$ $h_0:mu_d>0$ $h_a:mu_d = 0$
d. $\bigcirc$ $h_0:mu_1=mu_2$ $h_a:mu_1>mu_2$

Explanation:

Step1: Define null and alternative hypotheses

The null hypothesis ($H_0$) is a statement of no - effect or no difference. The alternative hypothesis ($H_a$) is what we are trying to find evidence for. Here, we want to test if first - born identical twins have higher IQ scores than second - born identical twins.
If $\mu_d$ is the mean of the differences (IQ score of first - born twin minus IQ score of second - born twin), the null hypothesis is that there is no difference, i.e., $\mu_d = 0$. The alternative hypothesis is that the first - born has a higher IQ, so $\mu_d>0$.

Step2: Analyze each option

• Option A: $H_0:\mu_d = 0$ and $H_a:\mu_d>0$ is correct as it represents no difference in the null and the claim of first - born having higher IQ in the alternative.
• Option B: $H_a:\mu_d < 0$ implies second - born have higher IQ, which is not what we want to test.
• Option C: The null and alternative are reversed from the correct form.
• Option D: While $\mu_1$ and $\mu_2$ are the means of first - born and second - born twins respectively, using $\mu_d$ is a more straightforward way for paired data. And the correct null and alternative in terms of $\mu_d$ are given in Option A.

Answer:

A. $H_0:\mu_d = 0$, $H_a:\mu_d>0$