Problem 28 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The number of Americans without health insurance, in millions, from 1995 through 2002 is summarized in the following table: $$\begin{array}{lllllllll}\hline \text { Year } & 1995 & 1996 & 1997 & 1998 & 1999 & 2000 & 2001 & 2002 \\ \hline \text { Number } & 40.7 & 41.8 & 43.5 & 44.5 & 40.2 & 39.9 & 41.2 & 43.6 \\\\\hline\end{array}$$ Find the average number of Americans without health insurance in the period from 1995 through 2002 . What is the standard deviation for these data?

Step-by-Step Solution

Answer

The average number of Americans without health insurance from 1995 to 2002 is approximately 41.925 million, with a standard deviation of about 1.62 million.

Step 1: Calculate the mean

To calculate the mean, we need to sum up all the values in the table and divide this sum by the total number of values (which is 8, one for each year from 1995 to 2002). Mean = \(\frac{\sum_{i=1}^{8} Number_i}{8}\) Mean = \(\frac{40.7+41.8+43.5+44.5+40.2+39.9+41.2+43.6}{8}\)

Step 2: Calculate the mean

Calculating the mean with the values given: Mean = \(\frac{335.4}{8}\) Mean ≈ 41.925 So the average number of Americans without health insurance from 1995 to 2002 is approximately 41.925 million.

Step 3: Calculate the squared differences

To find the standard deviation, we first need to calculate the squared differences between each data value and the mean. Squared differences = \((Number_i - Mean)^2\) For each year, the squared difference is as follows: Year 1995: \((40.7-41.925)^2 ≈ 1.50\) Year 1996: \((41.8-41.925)^2 ≈ 0.016\) Year 1997: \((43.5-41.925)^2 ≈ 2.46\) Year 1998: \((44.5-41.925)^2 ≈ 6.62\) Year 1999: \((40.2-41.925)^2 ≈ 2.98\) Year 2000: \((39.9-41.925)^2 ≈ 4.10\) Year 2001: \((41.2-41.925)^2 ≈ 0.526\) Year 2002: \((43.6-41.925)^2 ≈ 2.80\)

Step 4: Calculate the average of squared differences

Next, we need to calculate the average of these squared differences, which is the sum of the squared differences divided by the total number of values (8). Average of squared differences = \(\frac{\sum_{i=1}^{8}(Number_i - Mean)^2}{8}\) Average of squared differences ≈ \(\frac{1.50 + 0.016 + 2.46 + 6.62 + 2.98 + 4.10 + 0.526 + 2.80}{8}\) Average of squared differences ≈ \(\frac{20.99}{8}\) Average of squared differences ≈ 2.624

Step 5: Calculate the standard deviation

Finally, we need to calculate the square root of the average of squared differences to obtain the standard deviation. Standard deviation = \(\sqrt{Average \ of \ squared \ differences}\) Standard deviation ≈ \(\sqrt{2.624}\) Standard deviation ≈ 1.62 So the standard deviation for the number of Americans without health insurance from 1995 through 2002 is approximately 1.62 million.

Key Concepts

Mean Calculation

Calculating the mean is a fundamental step in data analysis; it represents the average value in a given set of data. To compute the mean, you sum up all the individual values and divide by the total number of data points. In practice, for a given dataset of numbers \( a_1, a_2, ..., a_n \), the mean \( \bar{x} \) is found using the formula:\[ \bar{x} = \frac{1}{n}\sum_{i=1}^{n} a_i \]
This calculation helps to find a simple central value for the dataset. For example, the mean number of Americans without health insurance from 1995 to 2002 serves to simplify and represent the overall trend during these years. By understanding the mean, you're better equipped to compare individual data points and understand the overall distribution of the data.

Squared Differences

The concept of squared differences plays a crucial role in measuring the spread or variability of a dataset. After finding the mean, we calculate the difference between each data point and the mean, then square this value. The formula for a squared difference of a single data point \( a_i \) is:\[ (a_i - \bar{x})^2 \]
Squaring serves a dual purpose: it eliminates negative values, which can skew the notion of spread, and it gives greater weight to outliers, or data points that are far from the mean. This calculation is a step toward finding the variance and ultimately the standard deviation, which quantifies how spread out the data points are from the mean.

Data Analysis

In data analysis, we collect, process, and interpret data to extract meaningful insights. By applying statistical measures like the mean and standard deviation, we can summarize complex datasets with a few descriptive numbers. The process often involves identifying trends, comparing subsets of data, and making predictions based on the data patterns. For educational purposes, fostering a clear understanding of data analysis fundamentals, like totaling the number of Americans without health insurance over several years, prepares students to tackle real-world problems where data interpretation is key.

Statistical Measures

Statistical measures are numerical descriptions of data's properties. The mean is a measure of central tendency, indicating the central or typical value in the dataset. Standard deviation, on the other hand, is a measure of dispersion that shows how much variation there is from the mean. These measures help compare across different datasets and are foundational concepts in fields ranging from economics to engineering. A firm grasp of these statistical tools not only aids in academic success but is also crucial in various professional contexts where data informed decisions are made.