Question

Belmont and Marolla conducted a study on the relationship between birth order, family size, and intelligence. The subjects consisted of all Dutch men who reached the age of 19 between 1963 and 1966. These men were required by law to take the Dutch army induction tests, including Raven’s intelligence test. The results showed that for each family size, measured intelligence decreased with birth order: first-borns did better than second-borns, second-borns did better than third-borns, and so on. And for any particular birth order, intelligence decreased with family size: for instance, first-borns in two-child families did better than firstborns in three-child families. Taking, for instance, men from two-child families:

• the first-borns averaged 2.58 on the test;

• the second-borns averaged 2.68 on the test.

(Raven test scores range from 1 to 6, with 1 being best and 6 worst.) The difference is small, but if it is real, it has interesting implications for genetic theory. To show that the difference was real, Belmont and Marolla made a two sample t-test. The standard deviation for the test scores was around one point in both groups, and there were 30,000 men in each group.

Belmont and Marolla concluded: “Thus the observed difference was highly significant . . .a high level of statistical confidence can be placed in each average because of the large number of cases.”

Do you agree with their conclusion? Why or why not? Was it appropriate to make a two-sample t-test in this situation?

Answer 1

μ₁: mean of Sample 1

µ₂: mean of Sample 2

Difference: μ₁ - µ₂

Equal variances are assumed for this analysis.

Descriptive Statistics

Sample	N	Mean	StDev	SE Mean
Sample 1	30000	2.58	1.00	0.0058
Sample 2	30000	2.68	1.00	0.0058

Estimation for Difference

Difference	Pooled StDev	95% CI for Difference
-0.10000	1.00000	(-0.11600, -0.08400)

Test

Null hypothesis	H₀: μ₁ - µ₂ = 0
Alternative hypothesis	H₁: μ₁ - µ₂ ≠ 0

T-Value	DF	P-Value
-12.25	59998	0.000

The p-value is 0.000.

Since the p-value (0.000) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that the observed difference was highly significant.

It is appropriate to make a two-sample t-test in this situation because we do not know the population standard deviation.

So, I agree with the conclusion.