91Ó°ÊÓ

Please do the following. (a) Draw a scatter diagram displaying the data. (b) Verify the given sums \(\Sigma x, \Sigma y, \Sigma x^{2}, \Sigma y^{2},\) and \(\Sigma x y\) and the value of the sample correlation coefficient \(r\) (c) Find \(\bar{x}, \bar{y}, a,\) and \(b .\) Then find the equation of the least- squares line \(\hat{y}=a+b x\) (d) Graph the least-squares line on your scatter diagram. Be sure to use the point \((\bar{x}, \bar{y})\) as one of the points on the line. (e) Interpretation Find the value of the coefficient of determination \(r^{2} .\) What percentage of the variation in \(y\) can be explained by the corresponding variation in \(x\) and the least-squares line? What percentage is unexplained? Answers may vary slightly due to rounding. Jobs An economist is studying the job market in Denver-area neighborhoods. Let \(x\) represent the total number of jobs in a given neighborhood, and let \(y\) represent the number of entry-level jobs in the same neighborhood. A sample of six Denver neighborhoods gave the following information (units in hundreds of jobs).Complete parts (a) through (e), given \(2 x=202,2 y=28, \Sigma x^{2}=7754\) \(\Sigma y^{2}=164, \Sigma x y=1096,\) and \(r \approx 0.860\) (f) For a neighborhood with \(x=40\) jobs, how many are predicted to be entry- level jobs?

Step by step solution

01

Draw the Scatter Diagram

To begin with, you need to plot the scatter diagram using the given data on a graph. The number of jobs in hundreds (x) will be plotted on the horizontal axis, and the number of entry-level jobs in hundreds (y) will be plotted on the vertical axis. This helps visually inspect the relationship between x and y.

02

Verify the Given Sums

Given the sums \(\Sigma x = 202\), \(\Sigma y = 28\), \(\Sigma x^{2} = 7754\), \(\Sigma y^{2} = 164\), and \(\Sigma xy = 1096\), ensure these values match with your dataset. To verify, manually sum all x values, y values, the squares of x, the squares of y, and the products of their pairs.

03

Calculate x̄, ȳ, a, and b

Calculate the means \(\bar{x} = \frac{\Sigma x}{n}\) and \(\bar{y} = \frac{\Sigma y}{n}\) where \(n=6\). Next, calculate the slope \(b\) of the least-squares line using the formula \(b = \frac{\Sigma xy - \frac{\Sigma x \Sigma y}{n}}{\Sigma x^2 - \frac{(\Sigma x)^2}{n}}\) and the intercept \(a = \bar{y} - b\bar{x}\).

04

Form the Equation of Least-Squares Line

Using the values of \(a\) and \(b\) from Step 3, formulate the equation of the least-squares line as \(\hat{y} = a + bx\).

05

Graph the Least-Squares Line

Add the least-squares line to your scatter plot. Ensure the line passes through the point \((\bar{x}, \bar{y})\), which acts as a pivot of the line on the graph.

06

Calculate and Interpret r²

Compute the coefficient of determination \(r^2 = r^2\). Given \(r \approx 0.860\), so \(r^2 \approx 0.7396\). This means about 73.96% of the variation in y is explained by x, while the remaining 26.04% is unexplained.

07

Predict Entry-Level Jobs for a Given x

Substitute \(x = 40\) into your equation of the least-squares line to find \(y\). Use \(\hat{y} = a + 40b\) to make the prediction.

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Key Concepts

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

Scatter Diagram

A scatter diagram is a vital tool in statistics used to visualize the relationship between two different variables. In this context, the diagram helps us understand how the total number of jobs in a neighborhood relates to the number of entry-level jobs. To create a scatter diagram:

• Draw two axes, with the horizontal axis representing the variable \(x\) (total jobs) and the vertical axis representing the variable \(y\) (entry-level jobs).
• Plot each neighborhood's data as a point on the graph where the \(x\) value is the total jobs and the \(y\) value is the entry-level jobs.

By examining the scatter diagram, one can roughly determine whether there is a positive correlation (where \(y\) increases as \(x\) increases) or a negative correlation (where \(y\) decreases as \(x\) increases). In our example, a scatter diagram suggests a positive correlation, helping to visually inspect relationships before delving into more complex calculations.

Least-Squares Line

The least-squares line, often known as the line of best fit, is a straight line that best expresses the relationship between the scatter points on your diagram. Its primary purpose is to provide an estimation or prediction, minimizing the distance (errors) between the line and all data points.Finding the least-squares line involves:

• Calculating the mean of \(x\), \(\bar{x}\), and the mean of \(y\), \(\bar{y}\).
• Determining the slope \(b\) and the intercept \(a\). The formula for the slope is \[ b = \frac{\Sigma xy - \frac{\Sigma x \Sigma y}{n}}{\Sigma x^2 - \frac{(\Sigma x)^2}{n}} \]
• The intercept is derived as \( a = \bar{y} - b\bar{x} \).

Once you have your slope and intercept, you can write the equation of the line as \( \hat{y} = a + bx \). This equation can predict \(y\) from any given \(x\), offering insights into how one might expect the number of entry-level jobs to behave as total jobs change.

Coefficient of Determination

Understanding how well the least-squares line describes the data is crucial. This is where the coefficient of determination, or \(r^2\), comes into play. It measures the proportion of the variation in \(y\) that can be predicted from \(x\) using our model.In simpler terms:

• \(r^2\) provides a percentage that indicates how much of the variance in \(y\) is accounted for by the variance in \(x\).
• For instance, an \(r^2\) value of 0.7396 suggests that approximately 73.96% of the variation in the number of entry-level jobs can be explained by the number of total jobs.

The remaining percentage, about 26.04% in our case, is considered unexplained or due to other factors not accounted for in the model. Thus, \(r^2\) is a helpful number, offering insights into the strength and reliability of the relationship between your variables.

Sample Correlation Coefficient

The sample correlation coefficient, represented by \(r\), quantifies the direction and strength of a linear relationship between two variables. It's a value between -1 and 1, where:

• 1 indicates a perfect positive linear relationship.
• -1 indicates a perfect negative linear relationship.
• 0 means there is no linear relationship.

In the case of our economist's study of Denver neighborhoods, the correlation coefficient \(r\) is approximately 0.860. This suggests a strong positive linear relationship between total jobs and entry-level jobs across different neighborhoods.To compute \(r\), you'd use the formula \[ r = \frac{n\Sigma xy - \Sigma x\Sigma y}{\sqrt{[n\Sigma x^{2} - (\Sigma x)^2][n\Sigma y^{2} - (\Sigma y)^2]}} \]This equation considers both individual variances and combined effects, offering a comprehensive view of the relationships inherent in the data. A strong correlation implies predictive capabilities, allowing for data-driven decisions.