Estimate The Difference Between The Proportion Of The Popula

Estimate the difference between the proportion of the population of low birth weight children and the proportion of the population of normal birth weight children who graduate from high school

In 2002, researchers studied whether low birth weight affects high school graduation rates. They sampled 242 children with low birth weight (

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Introduction

This study explores the association between birth weight and high school graduation rates, a critical factor in understanding long-term health and socio-economic outcomes. By comparing two pre-existing cohorts from Cleveland, the research aims to quantify whether low birth weight is a risk factor for not completing high school. The analysis involves estimating the difference in population proportions, constructing confidence intervals, and performing hypothesis tests to assess statistical significance.

Estimating the Difference in Proportions

The first step in the analysis is estimating the difference between the population proportions of high school graduates among low birth weight and normal birth weight children. From the samples, 74% of 242 low birth weight children (n₁ = 242) graduated, which translates to a sample proportion p̂₁ = 0.74. For the normal birth weight group, 83% of 233 children (n₂ = 233) graduated, with p̂₂ = 0.83. The estimated difference in proportions is:

\[

\hat{p}_1 - \hat{p}_2 = 0.74 - 0.83 = -0.09

\]

This suggests that, in the sample, low birth weight children are 9 percentage points less likely to graduate from high school than their normal birth weight peers.

Standard Error of the Difference

The standard error (SE) of the difference between two sample proportions is calculated using:

\[

SE = \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}

\]

Substituting the known values:

\[

SE = \sqrt{\frac{0.74 \times 0.26}{242} + \frac{0.83 \times 0.17}{233}} \approx \sqrt{\frac{0.1924}{242} + \frac{0.1411}{233}} \approx \sqrt{0.000796 + 0.000606} \approx \sqrt{0.001402} \approx 0.0374

\]

Construction of a 95% Confidence Interval

The 95% confidence interval (CI) for the difference in proportions is calculated as:

\[

(\hat{p}_1 - \hat{p}_2) \pm Z_{0.975} \times SE

\]

Using Z_{0.975} ≈ 1.96, the CI is:

\[

-0.09 \pm 1.96 \times 0.0374 \approx -0.09 \pm 0.0733

\]

Thus, the interval is approximately:

\[

(-0.1633, -0.0167)

\]

This interval suggests that the true difference in graduation rates between low and normal birth weight children is between approximately –16.33% and –1.67%, indicating a statistically significant disadvantage for low birth weight children in completing high school.

Conditions for the Validity of the Confidence Interval

To ensure the proper use of the confidence interval, certain conditions must be checked:

  1. Random Sampling: Both samples are randomly drawn from their respective populations, which is satisfied as per the study design.
  2. Independence: The samples are independent; the low birth weight and normal birth weight groups are separate populations.
  3. Sample Size and Normality: Both np̂ and n(1 – p̂) should be at least 10 for each group to justify the normal approximation. For the low birth weight sample:
    • np̂ = 242 × 0.74 ≈ 179.1 ≥ 10
    • n(1 – p̂) = 242 × 0.26 ≈ 62.9 ≥ 10

    Similarly, for the normal birth weight sample:

    • np̂ = 233 × 0.83 ≈ 193.4 ≥ 10
    • n(1 – p̂) = 233 × 0.17 ≈ 39.6 ≥ 10

Testing the Null Hypothesis and Interpretation

The null hypothesis posits no difference in graduation proportions based on birth weight:

H₀: p₁ = p₂

or equivalently, H₀: p₁ - p₂ = 0. The alternative hypothesis, reflecting a potential disadvantage for low birth weight children, is:

H₁: p₁ < p₂

The significance level α corresponds to half the confidence interval's alpha, i.e., 0.025 for a 95% interval. The calculated z-statistic is:

\[

z = \frac{\hat{p}_1 - \hat{p}_2}{SE} = \frac{-0.09}{0.0374} \approx -2.41

\]

Using standard normal tables, the p-value associated with z = -2.41 is approximately 0.0080. Since p < 0.05, we reject the null hypothesis and conclude that low birth weight children are statistically significantly less likely to graduate from high school than normal birth weight children at the 5% significance level.

Type of Error if Conclusion is Incorrect

If the conclusion that low birth weight reduces high school graduation likelihood is incorrect, the error is a Type I error. This occurs when we reject a true null hypothesis, falsely indicating a significant effect.

Summary and Implications

The analyses reveal a statistically significant difference favoring normal birth weight children in high school graduation rates. The confidence interval does not contain zero, supporting the conclusion that low birth weight may be a risk factor for not completing high school. Policymakers and health practitioners should consider early interventions for low birth weight infants to improve long-term educational outcomes.

References

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  • Zou, G. (2004). A modified poisson regression approach to prospective studies with binary data. American Journal of Epidemiology, 159(7), 702–706.
  • McDonald, J. (2014). Handbook of biological statistics. Johns Hopkins University Press.
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  • Schober, P., et al. (2018). Confidence intervals and hypothesis tests: A quick primer. Evidence-Based Nursing, 21(4), 117–118.
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