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Chapter 9: Problem 11

The level of lead in the blood was determined for a sample of 152 male hazardous-waste workers ages \(20-30\) and also for a sample of 86 female workers, resulting in a mean \(\pm\) standard error of \(5.5 \pm 0.3\) for the men and \(3.8 \pm 0.2\) for the women ("Temporal Changes in Blood Lead Levels of Hazardous Waste Workers in New Jersey, 19841987, Environ. Monitoring and Assessment, 1993: 99-107). Calculate an estimate of the difference between true average blood lead levels for male and female workers in a way that provides information about reliability and precision.

Short Answer

Expert verified
The estimated difference is 1.7 (0.9944, 2.4056) at 95% confidence.

Step by step solution

01

Identify Given Information

The mean blood lead level for male workers is \( \bar{x}_m = 5.5 \) with a standard error (SE) of \( 0.3 \). For female workers, the mean is \( \bar{x}_f = 3.8 \) with a standard error of \( 0.2 \). The sample sizes are \( n_m = 152 \) for males and \( n_f = 86 \) for females.
02

Calculate the Estimate of the Difference in Means

The point estimate for the difference in the average blood lead levels between male and female workers is calculated by subtracting the two means: \[ \bar{x}_m - \bar{x}_f = 5.5 - 3.8 = 1.7. \] This suggests that male workers have higher blood lead levels on average than female workers by 1.7 units.
03

Determine the Standard Error of the Difference

The standard error for the difference in sample means is computed using the formula: \( SE_{\text{diff}} = \sqrt{SE_m^2 + SE_f^2} \). Substituting the given values gives: \[ SE_{\text{diff}} = \sqrt{0.3^2 + 0.2^2} = \sqrt{0.09 + 0.04} = \sqrt{0.13} \approx 0.36. \]
04

Construct a Confidence Interval for the Difference

Assuming a normal distribution of blood lead levels, a confidence interval (CI) for the difference in population means can be constructed using \( \bar{x}_m - \bar{x}_f \pm z \times SE_{\text{diff}} \). For a 95% confidence level, \( z \approx 1.96 \). The interval is: \[ 1.7 \pm 1.96 \times 0.36 = 1.7 \pm 0.7056. \] Simplifying gives (0.9944, 2.4056).
05

Interpret the Results

The 95% CI for the difference in true average blood lead levels between male and female workers is (0.9944, 2.4056). This indicates we can be 95% confident that the true difference in average blood lead levels between genders falls within this interval, suggesting male workers have higher average blood lead levels.

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Key Concepts

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

Confidence Interval
A confidence interval (CI) is a range of values that is likely to contain the true difference in means between two groups. It gives us an idea of how precisely we can estimate this difference. For this exercise, we calculated the confidence interval using a known standard normal distribution.
The steps to construct a confidence interval include:
  • Calculating the point estimate of the difference.
  • Determining the standard error of the difference.
  • Using the appropriate z-value for the desired confidence level (in this case, 95% confidence corresponds to a z-value of 1.96).
By applying these steps, we find that the 95% confidence interval for the difference in true average blood lead levels between male and female workers is roughly (0.9944, 2.4056).
This means we are 95% confident that the actual difference in average blood lead levels between these two groups falls within this range.
Difference in Means
The difference in means tells us how much one group's mean measurement exceeds or falls short of another group's mean. In our example, we calculated this difference by subtracting the average blood lead level of female workers from that of male workers. This simple subtraction gives us the point estimate:
  • The mean for male workers: 5.5
  • The mean for female workers: 3.8
Thus, the difference in means is: \[\bar{x}_m - \bar{x}_f = 5.5 - 3.8 = 1.7\] This result suggests that, on average, male workers have higher blood lead levels than female workers by 1.7 units.
Determining this difference is crucial for understanding the magnitude of the disparity between the two groups.
Standard Error
The standard error (SE) measures the amount of variation or "spread" in the sample mean. It tells us how much the sample mean is expected to differ from the true population mean. In the context of this exercise, we computed the standard error of the difference in the blood lead levels for two groups.
Here's how you compute the standard error for the difference between two independent sample means:
  • Calculate the SE for each group: \( SE_m = 0.3 \) for males and \( SE_f = 0.2 \) for females.
  • Use the formula to find the SE of the difference: \[SE_{diff} = \sqrt{SE_m^2 + SE_f^2}\]
  • Substitute the values: \[SE_{diff} = \sqrt{0.3^2 + 0.2^2} = \sqrt{0.09 + 0.04} = \sqrt{0.13} \approx 0.36\]
The standard error of the difference, approximately 0.36, indicates the variability in the difference of blood lead levels between the two sample groups.
Point Estimate
A point estimate is a single value used to estimate a population parameter. It represents the best guess of the true parameter value based on sample data. In this scenario, the point estimate of interest is the difference in means between male and female blood lead levels.
To find the point estimate:
  • Subtract the mean blood lead level of female workers from the mean of male workers.
  • This difference, \(1.7\), serves as the point estimate for the difference between the two populations' blood lead levels.
Point estimates provide a straightforward way to compare groups, though they don't provide information about variability or precision, which is why they are often paired with confidence intervals.

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Most popular questions from this chapter

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