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Chapter 13: Problem 48

Can the values of \(B\) and \(\rho\) calculated for the same population data have different signs? Explain.

Short Answer

Expert verified
No, the signs of \(B\) and \(\rho\) cannot be different when calculated for the same population data because they are both determined by the direction of the relationship between the variables.

Step by step solution

01

Understand \(B\) and \(\rho\)

Understand that both the slope \(B\) and the correlation coefficient \(\rho\) have to do with the relationship between two variables in a data set. The slope \(B\) in a linear regression model is a measure of how much the dependent variable changes for a one-unit change in the independent variable. The correlation coefficient \(\rho\) is a measure of how strongly the two variables are related.
02

Understand the sign of the values

The sign of \(B\) is determined by the trend of the data points; if the dependent variable tends to increase as the independent variable does, \(B\) is positive. Similarly, an increase in one variable being associated with a decrease in the other corresponds to a negative sign of \(B\). The sign of \(\rho\), on the other hand, is determined by the direction of the relationship. A positive correlation means that as one variable increases, the other does too. A negative correlation means that as one variable increases, the other decreases.
03

Determine whether the signs can differ

Given that the signs of \(B\) and \(\rho\) both rely on the direction of the relationship between the two variables, it is not possible for them to have different signs. Both values will either be positive or negative, depending on the relationship of the variables in the data set.

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Most popular questions from this chapter

The following table gives information on the calorie count and grams of fat for 8 of the many types of bagels produced and sold by Panera Bread. $$ \begin{array}{lcc} \hline \text { Bagel } & \text { Calories } & \text { Fat (grams) } \\ \hline \text { Asiago Cheese } & 330 & 6.0 \\ \text { Blueberry } & 340 & 1.5 \\ \text { Cinnamon Crunch } & 420 & 6.0 \\ \text { Cinnamon Swirl \& Raisin } & 320 & 2.0 \\ \text { Everything } & 300 & 2.5 \\ \text { French Toast } & 350 & 4.0 \\ \text { Plain } & 290 & 1.5 \\ \text { Sesame } & 310 & 3.0 \\ \hline \end{array} $$ a. Find the least squares regression line with calories as the dependent variable and fat content as the independent variable. b. Make a 95\% confidence interval for \(B\). c. Test at the \(5 \%\) significance level whether \(B\) is different from \(14 .\)

Will you expect a positive, zero, or negative linear correlation between the two variables for each of the following examples? a. SAT scores and GPAs of students b. Stress level and blood pressure of individuals c. Amount of fertilizer used and yield of corn per acre d. Ages and prices of houses e. Heights of husbands and incomes of their wives

While browsing through the magazine rack at a bookstore, a statistician decides to examine the relationship between the price of a magazine and the percentage of the magazine space that contains advertisements. The data collected for eight magazines are given in the following table. $$ \begin{array}{l|rrrr} \hline \text { Percentage containing ads } & 37 & 43 & 58 & 49 \\ \hline \text { Price (\$) } & 5.50 & 6.95 & 4.95 & 5.75 \\ \hline \text { Percentage containing ads } & 70 & 28 & 65 & 32 \\ \hline \text { Price (\$) } & 3.95 & 8.25 & 5.50 & 6.75 \\ \hline \end{array} $$ a. Construct a scatter diagram for these data. Does the scatter diagram exhibit a linear relationship between the percentage of a magazine's space containing ads and the price of the magazine? b. Find the estimated regression equation of price on the percentage of space containing ads. c. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part b. d. Plot the estimated regression line on the scatter diagram of part a, and show the errors by drawing vertical lines between scatter points and the predictive regression line. e. Predict the price of a magazine with \(50 \%\) of its space containing ads. f. Estimate the price of a magazine with \(99 \%\) of its space containing ads. Comment on this finding.

The management of a supermarket wants to find if there is a relationship between the number of times a specific product is promoted on the intercom system in the store and the number of units of that product sold. To experiment, the management selected a product and promoted it on the intercom system for 7 days. The following table gives the number of times this product was promoted each day and the number of units sold. $$ \begin{array}{lc} \hline \begin{array}{c} \text { Number of Promotions } \\ \text { per Day } \end{array} & \begin{array}{c} \text { Number of Units Sold } \\ \text { per Day (hundreds) } \end{array} \\ \hline 15 & 11 \\ 22 & 22 \\ 42 & 30 \\ 30 & 26 \\ 18 & 17 \\ 12 & 15 \\ 38 & 23 \\ \hline \end{array} $$ a. With the number of promotions as an independent variable and the number of units sold as a dependent variable, what do you expect the sign of \(B\) in the regression line \(y=A+B x+\varepsilon\) will be? b. Find the least squares regression line \(\hat{y}=a+b x .\) Is the sign of \(b\) the same as you hypothesized for \(B\) in part a? c. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part b. d. Compute \(r\) and \(r^{2}\) and explain what they mean. e. Predict the number of units of this product sold on a day with 35 promotions. f. Compute the standard deviation of errors. g. Construct a \(98 \%\) confidence interval for \(B\). h. Testing at a \(1 \%\) significance level, can you conclude that \(B\) is positive? i. Using \(a=.02\), can you conclude that the correlation coefficient is different from zero?

A researcher took a sample of 10 years and found the following relationship between \(x\) and \(y\), where \(x\) is the number of major natural calamities (such as tornadoes, hurricanes, earthquakes, floods, etc.) that occurred during a year and \(y\) represents the average annual total profits (in millions of dollars) of a sample of insurance companies in the United States. $$ \hat{y}=342.6-2.10 x a. A randomly selected year had 24 major calamities. What are the expected average profits of U.S. insurance companies for that year? b. Suppose the number of major calamities was the same for each of 3 years. Do you expect the average profits for all U.S. insurance companies to be the same for each of these 3 years? Explain. c. Is the relationship between \(x\) and \(y\) exact or nonexact? $$
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