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The chapter introduction gave the accompanying data on the percentage of those eligible for a low-income subsidy who had signed up for a Medicare drug plan in each of 49 states (information was not available for Vermont) and the District of Columbia (USA Today, May 9 , 2006). \(\begin{array}{llllllll}24 & 27 & 12 & 38 & 21 & 26 & 23 & 33 \\ 19 & 19 & 26 & 28 & 16 & 21 & 28 & 20 \\ 21 & 41 & 22 & 16 & 29 & 26 & 22 & 16 \\ 27 & 22 & 19 & 22 & 22 & 22 & 30 & 20 \\ 21 & 34 & 26 & 20 & 25 & 19 & 17 & 21 \\ 27 & 19 & 27 & 34 & 20 & 30 & 20 & 21 \\ 14 & 18 & & & & & & \end{array}\) a. Compute the mean for this data set. b. The article stated that nationwide, \(24 \%\) of those eligible had signed up. Explain why the mean of this data set from Part (a) is not equal to 24 . (No information was available for Vermont, but that is not the reason that the mean differs-the \(24 \%\) was calculated excluding Vermont.)

Step by step solution

01

Calculate the mean

To calculate the mean or average of a data set, add up all the individual numbers in the set, then divide by the total count of numbers. So, sum up all the data points: 24, 27, 12,...,18 and divide by the total count, which is 49 states (excluding Vermont) plus the District of Columbia, for a total of 50.

02

Apply the formula and solve

Using the formula for mean \(\mu = \frac{\Sigma x_i}{n}\), where \(x_i\) are the individual data points and \(n\) is the number of data points, compute the mean.

03

Compare the calculated mean with 24%

Now, compare the calculated mean from part (a) with the 24% mentioned in the problem. If these two are not equal, it suggests that the coverage varies significantly among the different locations.

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Key Concepts

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

Data Analysis Essentials

Data analysis serves as a significant tool in understanding large sets of information. It involves collecting, cleaning, and interpreting data to uncover patterns and insights. In the problem provided, data analysis focuses on understanding the percentage of those eligible for a low-income subsidy who had signed up for a Medicare drug plan across different states and the District of Columbia.
By analyzing this data, we look for trends and deviations that could explain why some states may have more people signing up compared to others. For instance, while the national average or mean might show one picture, individual states can deviate based on numerous factors.

• The primary goal of data analysis here is to compute the mean which represents the central tendency of the data set.
• It is also crucial to bear in mind how each data point contributes to the overall outcome, such as variance indicating how much the data points differ from the mean.
• Data analysis helps identify these variations and characterize the dataset more comprehensively.

Understanding Descriptive Statistics

Descriptive statistics are summary statistics that quantitatively describe or summarize features from data. They provide a simple way to define the basic characteristics of a dataset without drawing any conclusions about the larger population from which the dataset was taken.
Key components include measures of central tendency like mean, median, and mode, and variability measures like range and standard deviation. For the problem at hand:

• Mean: Calculated by adding up all numbers and dividing by their count, it gives us the average percentage of sign-ups across states.
• Median: It is another measure showing the middle point of data when ordered from least to greatest, providing a sense of the entire data spread.
• Standard Deviation: Represents how much the individual percentages deviate from the mean, helping assess the spread or dispersion of the data.

Descriptive statistics allow us to present large amounts of data in a manageable form and help compare different datasets like the state percentages to the national 24% average.

Variance Explanation in Context

Variance is a statistical measure that describes how each number in a dataset differs from the mean. It helps illustrate the degree of variation or dispersion within a set of data points. In simple terms, while the mean gives a central value, variance shows how much the data points deviate from this central figure.
For the task at hand, if the variance is high, it means there is a significant fluctuation in the percentages of sign-ups among different states. Conversely, a low variance would imply that the sign-ups are more consistently around the mean value.

• The formula for variance in a sample dataset is \(s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}\), illustrating the average squared deviation from the mean.
• Understanding variance helps explain why the mean might not match a stated national average, highlighting discrepancies in population coverage based on regional variables.
• Variance provides insights into data consistency and reliability, offering a window into how different factors at the state level might be at play.

Therefore, variance is crucial when seeking to comprehend the dynamics of the data set in question.