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Let \(x\) be per capita income in thousands of dollars. Let \(y\) be the number of medical doctors per 10,000 residents. Six small cities in Oregon gave the following information about \(x\) and \(y\) (based on information from Life in America's Small Cities, by G. S. Thomas, Prometheus Books). $$ \begin{array}{c|cccccc} \hline x & 8.6 & 9.3 & 10.1 & 8.0 & 8.3 & 8.7 \\ \hline y & 9.6 & 18.5 & 20.9 & 10.2 & 11.4 & 13.1 \\ \hline \end{array} $$ Complete parts (a) through (e), given \(\Sigma x=53, \Sigma y=83.7, \Sigma x^{2}=471.04\), \(\Sigma y^{2}=1276.83, \Sigma x y=755.89\), and \(r \approx 0.934 .\) (f) Suppose a small city in Oregon has a per capita income of 10 thousand dollars. What is the predicted number of M.D.s per 10,000 residents?

Step by step solution

01

Understand the Problem

We need to predict the number of medical doctors per 10,000 residents based on a given per capita income for a city using a linear regression model. We are given data of per capita incomes and doctor numbers for six cities, along with certain statistical sums.

02

Recall the Regression Equation

The regression equation is of the form \[y = a + bx\]where \(a\) is the y-intercept and \(b\) is the slope of the line.

03

Calculate the Slope (b)

The slope \(b\) can be calculated using the formula:\[b = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{n(\Sigma x^2) - (\Sigma x)^2}\]where \(n\) is the number of data points. Here, \(n = 6\). Substituting the known values:\[b = \frac{6(755.89) - (53)(83.7)}{6(471.04) - (53)^2}\]

04

Calculate the Slope (b) Continued

Continue the calculation:\[b = \frac{4535.34 - 4436.1}{2826.24 - 2809} = \frac{99.24}{17.24} \approx 5.756\]

05

Calculate the Y-intercept (a)

The y-intercept \(a\) is calculated using:\[a = \frac{(\Sigma y) - b(\Sigma x)}{n}\]Substitute the known values:\[a = \frac{83.7 - (5.756)(53)}{6}\]

06

Calculate the Y-intercept (a) Continued

Continue the calculation:\[a = \frac{83.7 - 305.068}{6} = \frac{-221.368}{6} = -36.8947\]

07

Write the Regression Line Equation

Substitute \(a\) and \(b\) into the regression equation:\[y = -36.8947 + 5.756x\]

08

Predict for x = 10

Substitute \(x = 10\) (the given income) into the regression equation to find \(y\):\[y = -36.8947 + 5.756(10)\]

09

Calculate the Predicted y

Calculate the result:\[y = -36.8947 + 57.56= 20.6653\]Thus, the predicted number of M.D.s per 10,000 residents for a city with per capita income of 10 thousand dollars is approximately 20.67.

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Key Concepts

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

Linear Regression

Linear regression is a statistical method that models the relationship between a dependent variable and one or more independent variables. In our exercise, the dependent variable is the number of medical doctors per 10,000 residents, denoted as \( y \), and the independent variable is the per capita income in thousands of dollars, denoted as \( x \). The primary goal of linear regression is to find the line that best fits the observed data. This line is expressed through the regression equation \( y = a + bx \), where \( a \) is the y-intercept, and \( b \) is the slope.

• Simplifies complex data relationships into a straight line.
• Helps make predictions about the dependent variable when given new values of the independent variable.
• Commonly used in various fields like economics, biology, and social sciences.

Understanding this basic concept helps you analyze and predict outcomes by drawing conclusions from data patterns.

Slope Calculation

The slope \( b \) of the regression line indicates how much \( y \) changes for a one-unit change in \( x \). It quantifies the relationship between the variables. To calculate the slope \( b \), use the formula:\[b = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{n(\Sigma x^2) - (\Sigma x)^2}\]where \( n \) is the number of data points. For our specific data set:

• \( \Sigma xy \) is the sum of the product of each \( x \) and \( y \) value.
• \( \Sigma x \) is the sum of all \( x \) observations.
• \( \Sigma y \) is the sum of all \( y \) observations.
• \( \Sigma x^2 \) is the sum of the squares of all \( x \) observations.

By plugging in these values, as shown in the exercise, we calculated \( b \approx 5.756 \). This means for every increase of 1 (thousand dollars) in per capita income, the number of medical doctors per 10,000 residents increases by approximately 5.756. Understanding how to calculate the slope helps interpret the direction and strength of a relationship between variables.

Y-intercept Calculation

The y-intercept \( a \) of a linear regression line represents the value of \( y \) when \( x \) is zero. It is the point where the line crosses the y-axis. To find \( a \), use the formula:\[a = \frac{(\Sigma y) - b(\Sigma x)}{n}\]Substitute the calculated slope \( b \) and other known values from the data:

• This value helps to understand the baseline level of \( y \) when there is no contribution from the independent variable.
• The calculated y-intercept \( a = -36.8947 \) suggests a theoretical scenario.
• In real-world scenarios, the y-intercept often provides a hypothetical rather than a practical interpretation.

Even if \( x = 0 \) in our dataset doesn't make sense practically (as zero income isn’t realistic), the intercept is vital for constructing the regression line and predicting \( y \) for any \( x \). Knowing how to calculate the y-intercept ensures that you can accurately construct and apply the regression equation.