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The paper 'Relationship Between Blood Lead and Blood Pressure Among Whites and African Americans" (a technical report published by Tulane University School of Public Health and Tropical Medicine, 2000 ) gave summary quantities for blood lead level (in micrograms per deciliter) for a sample of whites and a sample of African Americans. Data consistent with the given summary quantities follow: \(\begin{array}{lrllrrrl}\text { Whites } & 8.3 & 0.9 & 2.9 & 5.6 & 5.8 & 5.4 & 1.2 \\ & 1.0 & 1.4 & 2.1 & 1.3 & 5.3 & 8.8 & 6.6 \\ & 5.2 & 3.0 & 2.9 & 2.7 & 6.7 & 3.2 & \\ \text { African } & 4.8 & 1.4 & 0.9 & 10.8 & 2.4 & 0.4 & 5.0 \\\ \text { Americans } & 5.4 & 6.1 & 2.9 & 5.0 & 2.1 & 7.5 & 3.4 \\ & 13.8 & 1.4 & 3.5 & 3.3 & 14.8 & 3.7 & \end{array}\) a. Compute the values of the mean and the median for blood lead level for the sample of African Americans. Which of the mean or the median is larger? What characteristic of the data set explains the relative values of the mean and the median? b. Construct a comparative boxplot for blood lead level for the two samples. Write a few sentences comparing the blood lead level distributions for the two samples.

Step by step solution

01

Calculating the Mean

To calculate the mean of blood lead level for the sample of African Americans, you will add up all the given values and then divide by the total number of values. Use this formula: \(Mean = \frac{Sum \: of \: Values}{Number \: of \: Values}\).

02

Calculating the Median

To calculate the median of blood lead level of the sample of African Americans, you will first arrange the values in ascending order, and then select the middle value if number of values is odd or the average of two middle values if number of values is even. The formula to find the median position when number of values is even: \(Median \: Position= \frac{Number \: of \: Values}{2} , \frac{Number \: of \: Values}{2} + 1\)

03

Comparing Mean and Median

Compare the calculated mean and median values. The larger value among these two indicates where the majority of data lies. A larger mean compared to median signifies that there are a few high values which are pulling the mean up (positive skew). Conversely, a larger median signifies lower values are pulling the mean down (negative skew).

04

Constructing a Boxplot

Boxplots help to visualize the complete range (min-max), the interquartile range (25% - 75%) and the median (50%) of data. For constructing comparative boxplots for blood lead levels for the two samples, software tools or graphing calculator that support statistics functions will be needed. Make sure to plot the median, lower quartile (Q1), upper quartile (Q3), minimum and maximum blood lead levels for each sample, and compare.

05

Comparing the Boxplots

Write a few sentences comparing the blood lead level distributions for the two samples based on the boxplots. Pay attention to comparisons of the medians, comparisons of the interquartile ranges (spread of the middle 50% data), comparisons of ranges (total spread of data), and presence of outliers.

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Key Concepts

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

Calculating Mean

The mean, or average, is a critical statistical measure that exemplifies the central tendency of a dataset. To calculate the mean blood lead level, as seen in step 1 of the solution, you simply sum all the values and divide by the number of observations. In mathematical terms, the formula for the mean \( \bar{x} \) is:
\[ \bar{x} = \frac{\sum{x_i}}{n} \]
where \( \sum{x_i} \) represents the sum of all data values and \( n \) is the number of data points. A larger mean can indicate the presence of higher values in the dataset, but it's important to keep in mind that the mean can be influenced by outliers, which are extreme values that differ significantly from other observations.

Calculating Median

The median is the middle value in a data set when ordered from smallest to largest. If there is an odd number of observations, the median is the central number. However, with an even number of observations, as in step 2 of the solution, the median is the average of the two central numbers. The formula for finding the position of the median when the number of observations is even is:
\[ Median \ Position = \frac{n}{2} , \frac{n}{2} + 1 \]
The median is a robust measure that is not as easily distorted by outliers or very large values, which is why it may be more representative of the 'typical' value in a skewed distribution.

Comparative Boxplot

A comparative boxplot displays the distribution of data across different categories side-by-side, making it easier to compare and contrast distributions. It highlights the median, quartiles, and possible outliers of each category. The typically shown elements include:

• Median (the middle line of the box)
• Quartiles (the edges of the box for Q1 and Q3)
• Whiskers (lines extending from the box to the minimum and maximum values)
• Possible outliers

When analyzing blood lead levels, as in step 4, constructing comparative boxplots of the two groups allows us to visually compare the spread and central tendency of each group's data. It also helps to identify any skewed distributions or outliers that could affect the mean.

Data Skewness

Skewness refers to the extent to which a data distribution differs from a symmetrical bell curve, or normal distribution, with equal tails on either side. In the process of comparing the mean and median, as in step 3, if the mean is higher than the median, this indicates a positive skew, meaning the data has a tail stretching towards higher values. Conversely, if the median is higher than the mean, this indicates a negative skew with a tail towards lower values. Data skewness is significant when examining blood lead levels because it highlights the presence of extremely high or low values in the dataset which can have profound implications for public health assessments and related policy-making.