Answer In Step-By-Step Form: Birth Weight, Gestation Age, Ho

Answer In Step By Step Formbirth Weight Kggestation Agehospital

Given the data on birth weight, gestation age, and hospital of birth, the problem involves constructing confidence intervals for the differences in mean gestation ages and birth weights between different hospitals, under varying assumptions about population variances and confidence levels.

Let's analyze and perform each step methodically.

Step 1: Summarize the Data

The data provided includes measurements of birth weight (kg), gestation age (weeks), and hospital of birth for multiple infants. The hospitals are labeled A, B, and C.

Identify the groups for comparison:

  • Hospital A
  • Hospital B
  • Hospital C

Extract the data for each hospital for gestation age and birth weight accordingly.

Step 2: Data Segregation and Calculations

Separate gestation ages for infants born in Hospitals B and C to compare their means in part (a). Similarly, obtain data for Hospitals A and B in parts (b) and (d) for the respective comparisons.

Calculations involve:

  • Sample means (\(\bar{x}\))
  • Sample variances (\(s^2\))
  • Sample sizes (\(n\))

Step 3: Constructing Confidence Intervals

Based on the assumptions about population variances (equal or not) and confidence levels (95% or 90%), employ the appropriate formulas:

a) 95% Confidence Interval for the difference in mean Gestation Age between Hospitals B and C, assuming equal variances

The formula for the confidence interval when variances are assumed equal is:

\[ CI = (\bar{x}_B - \bar{x}_C) \pm t_{(1 - \alpha/2, df)} \times SE \]

where

\[ SE = \sqrt{s_p^2 \left(\frac{1}{n_B} + \frac{1}{n_C}\right)} \]

and pooled variance \(s_p^2\) is computed from data.

b) 95% Confidence Interval for the difference in mean Gestation Age between Hospitals A and B, assuming unequal variances (Welch's t-test)

Formula:

\[ CI = (\bar{x}_A - \bar{x}_B) \pm t_{(*, df)} \times SE \]

with degrees of freedom calculated using Welch’s approximation and standard error:

\[ SE = \sqrt{\frac{s_A^2}{n_A} + \frac{s_B^2}{n_B}} \]

c) 90% Confidence Interval for the difference in mean Birth Weight between Hospitals A and C, assuming equal variances

Same as (a), but with 90% confidence level:

\[ CI = (\bar{x}_A - \bar{x}_C) \pm t_{(1 - \alpha/2, df)} \times SE \]

Step 4: Calculations and Results

Perform the actual calculations with the data, compute sample means, variances, pooled or individual variances as needed, find degrees of freedom, t-values, standard errors, and final intervals.

Note: Given the data in the problem, specific numerical calculations would follow. Since actual data isn't fully tabulated here, the process is outlined conceptually. In practice, one would compute each statistic using the sample data.

Step 5: Interpretation of Confidence Intervals

a) Interpretation for the interval in part (a)

The 95% confidence interval for the difference in mean gestation age between infants born in Hospitals B and C suggests that we are 95% confident that the true difference in population means lies within this interval. If the interval contains zero, it indicates no significant difference; if it does not, it suggests a significant difference in average gestation age between these hospitals.

b) Interpretation for the interval in part (d)

The 90% confidence interval for the difference in mean birth weight between Hospitals A and C provides a range of values within which the true mean difference likely falls. If this interval includes zero, it indicates no statistically significant difference; otherwise, it suggests a meaningful difference in birth weights between these hospitals.

Conclusion

This analysis highlights how statistical inference allows us to estimate and compare population parameters based on sample data. The assumptions about variances strongly influence the choice of method for constructing confidence intervals, emphasizing the importance of correctly assessing variance equality in practical analysis.

References

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