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Chapter 7: Problem 95

Another type of outcome reported was fetal deaths. Suppose the investigators expect, given general population pregnancy statistics at the time of the survey, that \(1.5 \%\) of pregnancies will result in a fetal death. They found that among 586 pregnancies where an exposed worker was the father, 11 resulted in a fetal death. Provide a \(95 \%\) Cl for the underlying fetal death rate among offspring of exposed men. Given the \(\mathrm{Cl}\), how do you interpret the results of the study?

Short Answer

Expert verified
The 95% CI for fetal death rate is 0.78% to 2.98%. It suggests no significant increase in risk compared to the general population's rate of 1.5%.

Step by step solution

01

Define the Parameters and the Formula

We are given that 11 fetal deaths occurred out of 586 pregnancies. Define the sample proportion \( \hat{p} \) as the number of fetal deaths divided by the total number of pregnancies: \( \hat{p} = \frac{11}{586} \). To calculate the \( 95\% \) confidence interval, we will use the formula for the confidence interval of a proportion: \[ \hat{p} \pm z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \] where \( z \) is the z-value corresponding to a \( 95\% \) confidence level, approximately 1.96, and \( n \) is the sample size.
02

Calculate the Sample Proportion

Calculate the sample proportion \( \hat{p} \): \( \hat{p} = \frac{11}{586} \approx 0.0188 \). This means that the observed proportion of fetal deaths in this sample is approximately \( 1.88\% \).
03

Compute the Standard Error

The standard error is given by \( \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \). Substituting the values: \[ \sqrt{\frac{0.0188 \times (1 - 0.0188)}{586}} \approx 0.0056 \]. This value represents the standard deviation of the sample proportion.
04

Determine the Margin of Error

The margin of error is \( z \times \text{(standard error)} \). Using \( z = 1.96 \), we have: \( 1.96 \times 0.0056 \approx 0.0110 \). This margin indicates the range above and below the sample proportion within which we expect the true proportion to lie, with \( 95\% \) confidence.
05

Calculate the Confidence Interval

Substitute \( \hat{p} \) and the margin of error into the confidence interval formula: \[ \hat{p} \pm 0.0110 \] means \[ 0.0188 \pm 0.0110 \], or \[ (0.0078, 0.0298) \]. Convert these to percentages: \( 0.78\% \) to \( 2.98\% \). This range is our \( 95\% \) confidence interval for the true fetal death rate.
06

Interpret the Confidence Interval

The \( 95\% \) confidence interval of \( 0.78\% \) to \( 2.98\% \) includes the \( 1.5\% \) expected rate, suggesting that the studied rate of fetal deaths among offspring of exposed workers is not significantly different from the general population rate. Hence, there is no strong evidence to suggest an increased risk because the expected rate is within the confidence interval.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fetal Death Rate
The fetal death rate is an important indicator when examining pregnancy outcomes. It represents the occurrence of fetal loss after a certain gestational age. In our context, fetal deaths are observed among pregnancies where the father is an exposed worker. We aim to identify if these pregnancies have a higher fetal death rate compared to the general population. A fetal death occurs due to numerous factors, sometimes linked to environmental or occupational exposures. Therefore, our study examines if there is any deviation in the fetal death rate among specific exposed groups from the norm. This is vital for understanding potential impacts on public health and designing interventions. By calculating confidence intervals, we determine if the observed fetal death rate significantly deviates from expected rates, such as the 1.5% assumed based on general statistics.
Study Interpretation
Interpretation of study results involves evaluating statistical findings in the context of pre-existing knowledge. In this example, we calculated a 95% confidence interval for fetal death rates with a range from 0.78% to 2.98%. This interval provides insights into the potential true fetal death rate among offspring where the father was an exposed worker. Since the expected rate of 1.5% falls within our confidence interval, it suggests no stark difference compared to the norm. Importantly, this interpretation indicates that there isn't significant statistical evidence to claim an increased fetal death risk in the studied group. Such insights guide future research focus and potential interventions, ensuring resources are allocated efficiently and effectively.
Proportion Calculation
Proportion calculation is a fundamental statistical procedure used to quantify the fraction of a total. Here, we start by calculating the sample proportion (\( \hat{p} \)) using the formula: \[ \hat{p} = \frac{11}{586} \].This calculation gives approximately 0.0188, which means 1.88%. It represents the observed rate of fetal deaths in our sample. Calculating the sample proportion is essential for constructing a confidence interval. It offers a snapshot of the direct findings but does not inherently include the margin for error or potential variability in other samples. Using proportion, alongside standard error and confidence levels, helps extend these findings to a broader context, allowing us to estimate the true underlying rate with a known degree of certainty.

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Most popular questions from this chapter

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A screening program for neuroblastoma (a type of cancer) was undertaken in Germany among children born between November \(1,1993,\) and June \(30,2000,\) who were 9 to 18 months of age between May 1995 and April 2000 [6]. A total of 1,475,773 children participated in the screening program, of whom 204 were diagnosed between 12 and 60 months of age. The researchers expected the incidence rate of neuroblastoma to be 7.3 per 100,000 children during this period in the absence of screening. Provide a \(95 \%\) Cl for the incidence rate of neuroblastoma in the screened population. Express the \(95 \%\) Cl as \(\left(p_{1}, p_{2}\right),\) where \(p_{1}\) and \(p_{2}\) are in the units of number of cases per 100,000 children. Is \(p_{0}(7.3 \text { cases per } 100,000\) children) in this interval?

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