Using The Data Below: Find The 99% Confidence Interval

Using The Data In Below Find The 99 Confidence Interval For The Varia

The data shown below represents the ages in months of 25 babies born at a particular hospital this year. Babies who are less than a month old are given an age of 0 months. Find the 99% confidence interval for the variance AND standard deviation of the ages of the babies born at this particular hospital this year.

The mean (average) X is given as X = Sum of all the ages of babies / Number of babies. X = 116/25 = 4.64 months. The standard deviation (SD) is 3.2259. The degree of freedom is d.f. = n – 1 = 24. To find the 99% confidence interval for the variance and standard deviation, we use the Chi-Square distribution.

Paper For Above Instruction

The purpose of this analysis is to determine the 99% confidence interval for the variance and standard deviation of ages of babies born in a particular hospital during the year. Using the summarized data provided—the mean age of 4.64 months, standard deviation of 3.2259, and sample size of 25—statistical methods within the Chi-Square distribution framework are employed to estimate these parameters. The analysis assumes that the ages are normally distributed, which is a standard assumption when dealing with small samples that rely on Chi-Square-based inference.

Understanding the variability in ages is crucial for healthcare planning and resource allocation within the hospital. Estimating the variance provides insights into how spread out the ages are, which may influence staffing schedules, neonatal care preparedness, and parental counseling. The confidence interval offers a range within which the true population variance (and consequently, the standard deviation) is expected to fall with a specified level of certainty—in this case, 99%.

Methodology for Calculating the Confidence Interval for Variance and Standard Deviation

The calculation begins with recognizing that the sample variance (s²) follows a chi-square distribution scaled by the degrees of freedom (df). The formulas for the confidence interval for the population variance (σ²) are as follows:

  • Lower limit: LC = ( (n - 1) * s² ) / χ²₁
  • Upper limit: UC = ( (n - 1) * s² ) / χ²₂

where χ²₁ and χ²₂ are the chi-square critical values corresponding to the lower and upper bounds, respectively, at the desired confidence level.

Given a 99% confidence level, the significance level α is 0.01, and α/2 = 0.005. The degrees of freedom are 24, which leads us to locate the relevant chi-square critical values from statistical tables:

  • χ²₀.005, 24 ≈ 39.36
  • χ²₀.995, 24 ≈ 12.40

The sample variance is calculated as s² = (standard deviation)² = 3.2259² ≈ 10.410.

Applying the formulas:

Lower limit for variance:

σ²₁ = ( (25 - 1) 10.410 ) / 39.36 ≈ (24 10.410) / 39.36 ≈ 249.84 / 39.36 ≈ 6.34

Upper limit for variance:

σ²₂ = (24 * 10.410) / 12.40 ≈ 249.84 / 12.40 ≈ 20.17

Therefore, the 99% confidence interval for the population variance is approximately (6.34, 20.17).

To find the confidence interval for the standard deviation (σ), take the square root of the variance limits:

  • Lower limit: √6.34 ≈ 2.52 months
  • Upper limit: √20.17 ≈ 4.49 months

Thus, the 99% confidence interval for the standard deviation of the ages of babies born in this hospital this year is approximately (2.52 months, 4.49 months).

This interval suggests that with 99% certainty, the true standard deviation of the ages falls within this range, highlighting the variability inherent in neonatal ages at this facility. The relatively wide range points to notable variability among the ages, which could be attributed to the diversity in birth timing, including very early or late births, and possibly some clinical factors influencing neonatal age at the time of assessment.

Conclusion

In summary, the analysis indicates that the population variance of baby ages is between approximately 6.34 and 20.17 months squared with 99% confidence. Correspondingly, the standard deviation is estimated to be between about 2.52 and 4.49 months. These estimates provide valuable insights for hospital administrators and healthcare providers in understanding and managing neonatal care resources and planning for future needs. The application of the Chi-Square distribution in this context underscores the importance of statistical methods in healthcare research and decision-making.

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