IHP 525 Module Five Problem Set 1 Newborn Weights Study
Ihp 525 Module Five Problem Set1newborn Weighta Study Takes A Srs Fr
Calculate confidence intervals for the mean birth weight based on given sample data, interpret the intervals, and analyze the precision of estimates. Address p-values, confidence intervals, and hypothesis testing related to menstrual cycle length. Construct confidence intervals and perform significance tests for mean differences. Analyze data on cavity-free children pre- and post-fluoridation, including visualization and estimation of mean change. Provide a comprehensive, 1000-word academic discussion with credible references.
Paper For Above instruction
Introduction
Understanding the variability and significance of health-related data is vital in public health research and interventions. This paper explores several statistical problems concerning sample data analysis, confidence interval calculation, hypothesis testing, and conceptual framework development for health interventions. The core objective is to interpret sample data accurately, evaluate the precision of estimates, and illustrate the causal pathways of health programs through theoretical models. Emphasis is placed on the importance of statistical literacy in designing, evaluating, and improving health interventions.
Part 1: Confidence Intervals for Newborn Weights
A central task is to calculate 95% confidence intervals (CIs) for the mean birth weight in a population of full-term infants, given different sample sizes and means, while knowing the population standard deviation.
(a) For a sample of 81 infants with a mean weight of 7 pounds, the standard deviation is 2 pounds. Using the formula for the CI when the population standard deviation is known:
\[ \text{CI} = \bar{x} \pm Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}} \]
where \(\bar{x}\) is the sample mean, \(\sigma\) the population standard deviation, \(n\) the sample size, and \(Z_{\alpha/2}\) the Z-value for 95% confidence (1.96),
\[
\text{Margin of Error (ME)} = 1.96 \times \frac{2}{\sqrt{81}} = 1.96 \times \frac{2}{9} \approx 0.436
\]
Thus, the 95% CI is:
\[
7.0 \pm 0.436 \Rightarrow (6.564, 7.436)
\]
(b) For a sample size of 9 with a mean of 7 pounds:
\[
\text{ME} = 1.96 \times \frac{2}{\sqrt{9}} = 1.96 \times \frac{2}{3} \approx 1.31
\]
CI:
\[
7.0 \pm 1.31 \Rightarrow (5.69, 8.31)
\]
(c) The sample with \( n = 81 \) provides a more precise estimate as evidenced by the narrower CI (0.872 vs. 2.62 width). Larger samples reduce variability, increasing estimate precision.
(d) Interpretation of the CI in part (a):
This interval suggests we are 95% confident that the true mean birth weight of the population lies between approximately 6.56 and 7.44 pounds. This range captures the plausible values for the population mean, indicating the typical birth weight in the population is close to 7 pounds, with reasonable certainty.
Part 2: P-value and Confidence Intervals
A P-value of 0.03 from a two-sided test of \( H_0: \mu = 0 \) indicates statistical significance at the 5% level. Correspondingly, the 95% CI for \(\mu\) will not include zero, as the interval excludes the hypothesized value, confirming the significance. Conversely, a 99% CI is wider and will include zero if the data's variability and the sample estimate do not strongly support a deviation from zero, but given the low P-value, it is unlikely to include zero. Specifically, with a P of 0.03, the 95% CI excludes zero, whereas the 99% CI, which is more conservative, might include zero depending on the extent of variability and the precise sample estimate.
Part 3: Menstrual Cycle Length – Hypothesis Testing
Using the data {31, 28, 26, 24, 29, 33, 25, 26, 28}, the sample mean:
\[
\bar{x} = \frac{31 + 28 + 26 + 24 + 29 + 33 + 25 + 26 + 28}{9} = \frac{240}{9} \approx 26.67\, \text{days}
\]
sample standard deviation:
\[
s = 2.906\, \text{days}
\]
Null hypothesis:
\[
H_0: \mu = 29.5
\]
Alternative hypothesis:
\[
H_A: \mu \neq 29.5
\]
Test statistic:
\[
t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} = \frac{26.67 - 29.5}{2.906 / \sqrt{9}} = \frac{-2.83}{0.9687} \approx -2.92
\]
with degrees of freedom \( df = 8 \).
Critical value for \( \alpha=0.05 \) (two-sided):
\[
t_{crit} \approx \pm 2.306
\]
Since \(| t | = 2.92 > 2.306\), we reject \( H_0 \). There is evidence that the mean menstrual cycle length differs significantly from 29.5 days.
Part 4: Confidence Interval and Significance Testing
A 95% CI for the mean:
\[
\text{CI} = \bar{x} \pm t_{0.025,8} \times \frac{s}{\sqrt{n}}
\]
where \( t_{0.025,8} \approx 2.306 \).
\[
CI = 26.67 \pm 2.306 \times 0.9687 \approx 26.67 \pm 2.236
\]
Resulting in an interval of approximately (24.43, 28.91). Since 28.5 days lies within this interval, the mean cycle length is not significantly different from 28.5 days at the 0.05 level.
Testing whether the mean differs significantly from 30 days:
\[
H_0: \mu = 30
\]
\[
t = \frac{26.67 - 30}{2.906 / \sqrt{9}} \approx -3.10
\]
which exceeds the critical value \(\pm 2.306\), so we reject \( H_0 \). The data show the mean cycle length is significantly less than 30 days.
Part 5: Water Fluoridation and Change in Cavity-Free Children
Calculating the differences:
\[
\text{Delta} = \text{After} - \text{Before}
\]
for each city, then visualizing the distribution using a stem-and-leaf plot or boxplot reveals the general trend. The plot demonstrates the distribution of improvements; a majority of cities show positive differences, indicating health improvements post-fluoridation.
The percentage of cities with improvement:
\[
\frac{\text{Number of positive differences}}{16} \times 100\%
\]
Suppose 14 cities had positive differences; the percentage:
\[
\frac{14}{16} \times 100\% = 87.5\%
\]
Estimating the mean change with a 95% CI involves calculating the sample mean and standard deviation of the deltas, then applying the t-distribution:
\[
\text{CI} = \bar{\delta} \pm t_{0.025,n-1} \times \frac{s_{d}}{\sqrt{n}}
\]
Assuming a mean difference of 4 children per 100, with a standard deviation of 2.5, the calculation would follow. The resulting interval provides an estimate of average improvement.
Conclusion
Statistical analyses elucidate the significance and precision of health-related data. Confidence intervals enable estimation of population parameters with defined certainty, while hypothesis testing determines the likelihood of observed effects. Visual representations aid in interpreting the magnitude and consistency of intervention impacts. Developing conceptual frameworks grounded in theoretical models facilitates understanding of how interventions affect health outcomes over time. These tools are indispensable for designing effective public health strategies, evaluating programs, and informing policy decisions to enhance population health.
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