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Chapter 7: Problem 17

A random sample of 860 births in New York State included 426 boys. Construct a 95\% confidence interval estimate of the proportion of boys in all births. It is believed that among all births, the proportion of boys is \(0.512 .\) Do these sample results provide strong evidence against that belief?

Short Answer

Expert verified
The 95\% confidence interval is (0.462, 0.528). These results do not strongly contradict the belief that the proportion of boys is 0.512.

Step by step solution

01

Identify the Sample Proportion

First, determine the sample proportion of boys in the sample. The sample proportion (\(\hat{p}\)) is calculated by dividing the number of boys by the total number of births. Therefore, \(\hat{p} = \frac{426}{860}\).
02

Calculate \(\hat{p}\)

Simplify the fraction to get the sample proportion: \(\hat{p} = \frac{426}{860} \approx 0.495.\)
03

Find the Standard Error

The standard error (SE) of the sample proportion is calculated using \(SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), where \(n\) is the sample size. Therefore, \(SE = \sqrt{\frac{0.495(1-0.495)}{860}} \approx 0.017.\)
04

Determine the Z-value for a 95\textbackslash\textbackslash% Confidence Interval

For a 95\% confidence interval, the critical value \(Z\) is approximately 1.96.
05

Calculate the Margin of Error

The margin of error (ME) is found by multiplying the Z-value by the standard error: \(ME = Z \times SE = 1.96 \times 0.017 \approx 0.033.\)
06

Construct the Confidence Interval

The confidence interval is calculated as \(\hat{p} \pm ME.\) So, the 95\% confidence interval is \(0.495 \pm 0.033\) or \( (0.462, 0.528).\)
07

Interpret the Results

The confidence interval (0.462, 0.528) contains the purported proportion of boys (0.512). Therefore, these sample results do not provide strong evidence against the belief that the proportion of boys is 0.512.

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Key Concepts

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

Sample Proportion
The sample proportion of boys in the sample simply means the percentage of all babies born in this sample group who are boys. To find it, we take the number of boys (426) and divide that by the total number of births (860).
This fraction \(\frac{426}{860}\) simplifies to approximately 0.495.
This value is our \(\hat{p}\), which is used in many of our subsequent calculations.
Sample proportions are essential because they help us estimate the population proportion, giving us a sense of what is happening in the entire population (all births in New York State in this case) based on our smaller sample.
Standard Error
Standard error (SE) measures the variability or the precision of the sample proportion. It's a way to see how much the sample proportion (0.495) might deviate from the true population proportion.
We calculate it with the formula: \(SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), where \(\hat{p}\) is the sample proportion and \(n\) is the sample size.
Plugging in our numbers gives us \(SE = \sqrt{\frac{0.495(1-0.495)}{860}} \approx 0.017\).
A small SE indicates that our sample proportion is likely very close to the true population proportion.
Margin of Error
The margin of error (ME) reflects the range within which the true population proportion likely falls.
For a 95% confidence interval, the critical value (Z) is approximately 1.96.
We find the margin of error with the formula: \(ME = Z \times SE = 1.96 \times 0.017 \approx 0.033.\)
This value, 0.033, tells us how much our sample proportion might vary from the actual population proportion.
So, if our sample proportion is 0.495, the true population proportion could reasonably be between 0.462 and 0.528.
Proportion Hypothesis Testing
Proportion hypothesis testing helps determine if our sample result provides strong evidence against a stated belief (null hypothesis).
The initial belief here is that the proportion of boys among all births is 0.512.
We compare this hypothesis to our sample result using our confidence interval of (0.462, 0.528).
Because 0.512 falls within our confidence interval, we do not have sufficient evidence to reject the belief.
This step is crucial because it uses statistical methods to determine the credibility of claims based on sample data.
Statistical Inference
Statistical inference allows us to make conclusions about a population based on sample data.
By calculating the sample proportion, standard error, margin of error, and confidence intervals, we infer characteristics of the entire population.
For our data, we infer that the proportion of boys in all births likely falls between 0.462 and 0.528 with 95% confidence.
This process transforms our specific sample findings into general understanding, giving decisions and predictions a scientific basis.
With these inferences, policymakers and healthcare professionals can better understand birth ratios and plan accordingly.

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Most popular questions from this chapter

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