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Chapter 13: Problem 24

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?

Short Answer

Expert verified
a. Expected profits for the year with 24 major calamities: $292.6 \t b. Yes, if other conditions remain constant and the number of calamities is the same, the profits should essentially be the same for all 3 years. c. The relationship between \(x\) and \(y\) is nonexact due to the nature of real-world data and because we are using a statistical model.

Step by step solution

01

Calculate Expected Profits

First, there's a need to substitute \(x=24\) into the regression equation to compute the expected profit. Using the linear regression model, \(\hat{y} = 342.6 - 2.1x\). By substitifying \(x = 24\), we get \(\hat{y} = 342.6 - 2.1(24)\). The equation simplification leads to the expected average profits.
02

Discuss Predictions for Multiple Years

Assuming that the number of major calamities remains constant over the years, if the relationship is perfectly linear and other factors are constant, the expected average profits for all U.S. insurance companies should essentially remain the same. This is because the calculation is based on the number of calamities, which in this case, does not change over time.
03

Assess Relationship Type

The relationship between \(x\) and \(y\) in this case is non-exact. Firstly, because in real world data, it's highly unlikely that variables are perfectly related, and secondly, because we are dealing with a statistical model here which is a simplification of reality and is based on estimates using sample data.

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Most popular questions from this chapter

A sample data set produced the following information. $$ \begin{aligned} &n=10, \quad \Sigma x=100, \quad \Sigma y=220, \quad \Sigma x y=3680 \\ &\Sigma x^{2}=1140, \text { and } \Sigma y^{2}=25,272 \end{aligned} $$ a. Calculate the linear correlation coefficient \(r\). b. Using the \(2 \%\) significance level, can you conclude that \(\rho\) is different from zero?

The following information is obtained from a sample data set. $$ \begin{aligned} &n=10, \quad \Sigma x=100, \quad \Sigma y=220, \quad \Sigma x y=3680 \\ &\Sigma x^{2}=1140, \text { and } \Sigma y^{2}=25,272 \end{aligned} $$ Find the values of \(s_{e}\) and \(r^{2}\).

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