Question

An investigator wants to show that first-born children have better vocabularies for their age than second-borns. He takes a simple random sample of 400 two-child families in a school district, both children being enrolled in elementary school. He gives these children the WISC vocabulary test, with the following results:

The 400 first-borns average 29 and their SD is 10.

The 400 second-borns average 27 and their SD is 10.

(Note: The scores are adjusted for age differences.) He makes a two-sample z-test, finding

SE for first born sample average = 0.5

SE for second born sample average = 0.5

SE for difference = 0.707

Z = (29-27)/0.707 = 2.83

P-value = 0.22%

Do you agree with the researcher’s conclusion that first-born children have better vocabularies for their age than second-borns? Why or why not?


Answer 1

Here, the hypotheses are:

H0 : The mean vocabulary test scores are equal for both groups

H1 : The first borns have a better mean vocabulary test score

We take alpha = 0.05

As we can see from the results, the p-value obtained is 0.22% , i.e , p-value = 0.002.

Hence, p-value < alpha. Hence, we reject the null hypothesis at alpha = 0.05 and we conclude that first born children have better vocabularies than second-borns.

Answer 2

The investigator concluded that first-born children have better vocabularies than second-borns based on a two-sample z-test, which yielded a statistically significant result (Z = 2.83, p-value = 0.22%). Here’s whether I agree and why:

Yes, the conclusion is statistically valid, but with caveats:

1. Statistical Significance: The p-value is very small (0.22%), meaning there’s strong evidence that the difference in vocabulary scores (29 vs. 27) is not due to random chance.

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3. Large Sample Size: With 400 families, the results are likely reliable, and the standard errors (SEs) are small, reinforcing confidence in the findings.

But consider these real-world factors:

• Practical Significance: While the difference is statistically significant, a 2-point gap (29 vs. 27) on the WISC vocabulary test may not be meaningful in everyday life. Is this small difference educationally or developmentally important?

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• Confounding Variables: The study doesn’t account for factors like parenting style, socioeconomic status, or school environment, which could influence vocabulary.

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• Correlation ≠ Causation: Even if first-borns score higher, it doesn’t prove birth order causes better vocabulary—other explanations (e.g., more parental attention early on) might apply.