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Chapter 13: Problem 17

A population data set produced the following information. $$ N=250, \quad \Sigma x=9880, \quad \Sigma y=1456, \quad \Sigma x y=85,080, \quad \Sigma x^{2}=485,870 $$ Find the population regression line.

Short Answer

Expert verified
The population regression line is obtained by substituting the computed values of 'a' and 'b' into the equation \(y = ax + b\). This gives the relationship between the dependent variable y and independent variable x.

Step by step solution

01

Calculate Slope (a)

Start by calculating the slope 'a' which can be calculated from the formulas \((n * \Sigma xy - \Sigma x * \Sigma y) / (n * \Sigma x^{2} - (\Sigma x)^{2})\). Substitute the given values into the formula: \(a = (250 * 85080 - 9880 * 1456) / (250 * 485870 - 9880^{2})\).
02

Calculate y-intercept (b)

Next, calculate the y-intercept 'b'. The formula is given as \((\Sigma y - a * \Sigma x) / n\). Substituting the slope and other given values into the formula will result in: \(b = (1456 - a * 9880) / 250\).
03

Determine Regression Line

Finally, to determine the regression line substitute values of 'a' and 'b' into the equation \[y = ax + b\]. This will give you the population regression line.

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Most popular questions from this chapter

For a sample data set, the slope \(b\) of the regression line has a negative value. Which of the following is true about the linear correlation coefficient \(r\) calculated for the same sample data? a. The value of \(r\) will be positive. b. The value of \(r\) will be negative. c. The value of \(r\) can be positive or negative.

The following information is obtained from a sample data set. $$ n=12, \quad \Sigma x=66, \quad \Sigma y=588, \quad \Sigma x y=2244, \quad \Sigma x^{2}=396 $$ Find the estimated regression line.

Plot the following straight lines. Give the values of the \(y\) -intercept and slope for each of these lines and interpret them. Indicate whether each of the lines gives a positive or a negative relationship between \(x\) and \(y\) a. \(y=100+5 x \quad\) b. \(y=400-4 x\)

The health department of a large city has developed an air pollution index that measures the level of several air pollutants that cause respiratory distress in humans. The accompanying table gives the pollution index (on a scale of 1 to 10 , with 10 being the worst) for 7 randomly selected summer days and the number of patients with acute respiratory problems admitted to the emergency rooms of the city's hospitals. $$ \begin{array}{l|ccccccc} \hline \text { Air pollution index } & 4.5 & 6.7 & 8.2 & 5.0 & 4.6 & 6.1 & 3.0 \\\ \hline \text { Emergency admissions } & 53 & 82 & 102 & 60 & 39 & 42 & 27 \\ \hline \end{array} $$ a. Taking the air pollution index as an independent variable and the number of emergency admissions as a dependent variable, do you expect \(B\) to be positive or negative in the regression model \(y=A+B x+\epsilon ?\) b. Find the least squares regression line. Is the sign of \(b\) the same as you hypothesized for \(B\) in part a? c. Compute \(r\) and \(r^{2}\), and explain what they mean. d. Compute the standard deviation of errors. e. Construct a \(90 \%\) confidence interval for \(B\). f. Test at the \(5 \%\) significance level whether \(B\) is positive. g. Test at the \(5 \%\) significance level whether \(\rho\) is positive. Is your conclusion the same as in part \(\mathrm{f}\) ?

Explain the meaning and concept of SSE. You may use a graph for illustration purposes.
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