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Chapter 13: Problem 72

Refer to Exercise \(13.25\). The data on ages (in years) and prices (in hundreds of dollars) for eight cars of a specific model are reproduced from that exercise. $$ \begin{array}{l|rrrrrrrr} \hline \text { Age } & 8 & 3 & 6 & 9 & 2 & 5 & 6 & 3 \\ \hline \text { Price } & 45 & 210 & 100 & 33 & 267 & 134 & 109 & 235 \\ \hline \end{array} $$ a. Do you expect the ages and prices of cars to be positively or negatively related? Explain. b. Calculate the linear correlation coefficient. c. Test at the \(2.5 \%\) significance level whether \(\rho\) is negative.

Short Answer

Expert verified
The ages and prices of cars are expected to be negatively related. The linear correlation coefficient can be calculated using deviations from the mean, multiplication of corresponding deviations, and standard deviation product. To test if \(\rho\) is negative, we use a one-tailed test with a null hypothesis that \(\rho = 0\), and an alternative that \(\rho < 0\). If our calculated \(\rho\) is significantly less than 0, then we conclude it is negative at a 2.5% significance level.

Step by step solution

01

Expectation

In general practice, It is typically expected that the age and price of a car are negatively related, meaning that as the age of a car increases, its price decreases. This is mainly because older cars commonly have more wear and tear, and higher mileage, thus reducing their value.
02

Calculation of linear correlation coefficient

The calculation is divided into three parts: (1) compute the deviations of the ages from their mean and the deviations of the prices from their mean; (2) Multiply corresponding deviations and add up the results (this is the covariance); (3) divide the result of (2) by the product of the standard deviation of ages and prices. These steps will give the linear correlation coefficient or Pearson鈥檚 r.
03

Hypothesis testing

Now, we want to test whether \(\rho\) is negative at a 2.5% significance level. We will use a one-tailed test for this purpose, because we are only concerned about whether \(\rho\) is less than zero, not whether it is greater than zero. Our null hypothesis is that \(\rho = 0\) (meaning the age and price of a car are not correlated), and our alternative hypothesis is that \(\rho < 0\). If the calculated correlation coefficient is significantly less than 0, we will reject the null hypothesis and accept the alternative hypothesis.

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Key Concepts

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

Negative Correlation
When we talk about negative correlation, we're looking at a situation where two variables move in opposite directions. For example, in this exercise, there is an expectation of negative correlation between the age and price of cars.
  • When the age of a car increases, we usually see its price decrease.
  • This is because older cars tend to have more wear and tear, which reduces their market value.
  • The concept of negative correlation can be applied in many real-world scenarios, such as temperature and heating costs in a home during winter.
Understanding negative correlation helps in predicting trends and making informed decisions. It signals that two variables have an inverse relationship, and this can influence strategies in business, finance, and even personal decisions.
Hypothesis Testing
Hypothesis testing is a method used to determine if there is enough statistical evidence to support a certain belief or hypothesis about a data set.In this problem, hypothesis testing is used to assess the relationship between car age and price.
  • Null Hypothesis (H0): The correlation coefficient (\( \rho \)) is equal to zero, indicating no correlation.
  • Alternative Hypothesis (H1): The correlation coefficient (\( \rho \)) is less than zero, indicating a negative correlation.
  • We use a 2.5% significance level for this test, meaning there鈥檚 a 2.5% chance of rejecting the null hypothesis incorrectly.
By analyzing the correlation coefficient calculated from the data, one can decide whether to reject the null hypothesis and conclude that there is a strong enough negative correlation. This process involves comparing the test statistic to a critical value or using a p-value approach.
Pearson鈥檚 r
Pearson鈥檚 r is the statistical measure that calculates the strength and direction of a linear relationship between two variables. It is also known as the linear correlation coefficient. In the car age and price problem:
  • To compute Pearson鈥檚 r, you first calculate the mean of both variables, the ages and prices in this case.
  • Then, find the deviations of each value from their respective means.
  • Next, multiply these corresponding deviations together to find the covariance.
  • Finally, Pearson鈥檚 r is determined by dividing the covariance by the product of the standard deviations of each variable.
Pearson鈥檚 r ranges from -1 to 1: - A value of -1 indicates a perfect negative linear relationship. - A value of 0 indicates no linear relationship. - A value of 1 indicates a perfect positive linear relationship. Calculating and interpreting Pearson鈥檚 r helps us understand if two variables, like age and price in cars, have a linear relationship and the strength of that relationship.

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Most popular questions from this chapter

Suppose that you work part-time at a bowling alley that is open daily from noon to midnight. Although business is usually slow from noon to 6 P.M., the owner has noticed that it is better on hotter days during the summer, perhaps because the premises are comfortably air-conditioned. The owner shows you some data that she gathered last summer. This data set includes the maximum temperature and the number of lines bowled between noon and 6 P.M. for each of 20 days. (The maximum temperatures ranged from \(77^{\circ} \mathrm{F}\) to \(95^{\circ} \mathrm{F}\) during this period.) The owner would like to know if she can estimate tomorrow's business from noon to 6 P.M. by looking at tomorrow's weather forecast. She asks you to analyze the data. Let \(x\) be the maximum temperature for a day and \(y\) the number of lines bowled between noon and 6 P.M. on that day. The computer output based on the data for 20 days provided the following results: \(\hat{y}=-432+7.7 x, \quad s_{e}=28.17, \quad \mathrm{SS}_{x x}=607, \quad\) and \(\quad \bar{x}=87.5\) Assume that the weather forecasts are reasonably accurate. a. Does the maximum temperature seem to be a useful predictor of bowling activity between noon and 6 P.M.? Use an appropriate statistical procedure based on the information given. Use \(\alpha=.05\) b. The owner wants to know how many lines of bowling she can expect, on average, for days with a maximum temperature of \(90^{\circ}\). Answer using a \(95 \%\) confidence level. c. The owner has seen tomorrow's weather forecast, which predicts a high of \(90^{\circ} \mathrm{F}\). About how many lines of bowling can she expect? Answer using a \(95 \%\) confidence level. d. Give a brief commonsense explanation to the owner for the difference in the interval estimates of parts \(\mathrm{b}\) and \(\mathrm{c}\). e. The owner asks you how many lines of bowling she could expect if the high temperature were \(100^{\circ} \mathrm{F}\). Give a point estimate, together with an appropriate warning to the owner.

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

The following table gives the total 2008 payroll (on the opening day of the season, rounded to the nearest million dollars) and the number of runs scored during the 2008 season by each of the National League baseball teams. $$ \begin{array}{lcc} \hline \text { Team } & \begin{array}{c} \text { Total Payroll } \\ \text { (millions of dollars) } \end{array} & \text { Runs Scored } \\ \hline \text { Arizona Diamondbacks } & 74 & 720 \\ \text { Atlanta Braves } & 97 & 753 \\ \text { Chicago Cubs } & 135 & 855 \\ \text { Cincinnati Reds } & 71 & 704 \\ \text { Colorado Rockies } & 75 & 747 \\ \text { Florida Marlins } & 37 & 770 \\ \text { Houston Astros } & 103 & 712 \\ \text { Los Angeles Dodgers } & 100 & 700 \\ \text { Milwaukee Brewers } & 80 & 750 \\ \text { New York Mets } & 136 & 799 \\ \text { Philadelphia Phillies } & 113 & 799 \\ \text { Pittsburgh Pirates } & 49 & 735 \\ \text { San Diego Padres } & 43 & 637 \\ \text { San Francisco Giants } & 82 & 640 \\ \text { St. Louis Cardinals } & 89 & 779 \\ \text { Washington Nationals } & 59 & 641 \\ \hline \end{array} $$ a. Find the least squares regression line with total payroll as the independent variable and runs scored as the dependent variable. b. Is the equation of the regression line obtained in part a the population regression line? Why or why not? Do the values of the \(y\) -intercept and the slope of the regression line give \(A\) and \(B\) or \(a\) and \(b\) ? c. Give a brief interpretation of the values of the \(y\) -intercept and the slope obtained in part a. d. Predict the number of runs scored by a team with a total payroll of \(\$ 84\) million.

The following table provides information on the high temperature for each day and the number of crimes committed in Chicago, Illinois, during the period July 1, 2009 to July 14, 2009. $$ \begin{array}{l|rrrrrrr} \hline \text { High temperature }\left({ }^{\circ} \mathrm{F}\right) & 65 & 73 & 79 & 69 & 81 & 86 & 77 \\ \hline \text { Number of crimes } & 1110 & 1134 & 1117 & 1044 & 1014 & 1105 & 1152 \\ \hline \text { High temperature }\left({ }^{\circ} \mathrm{F}\right) & 65 & 79 & 82 & 85 & 82 & 79 & 80 \\ \hline \text { Number of crimes } & 1046 & 1127 & 1160 & 1065 & 1126 & 1041 & 1038 \\ \hline \end{array} $$ a. Find the least squares regression line \(\hat{y}=a+b x\). Take high temperature as an independent variable and number of crimes committed as a dependent variable. b. Give a brief interpretation of the values of \(a\) and \(b\). c. Compute \(r\) and \(r^{2}\) and explain what they mean. d. Predict the number of crimes committed on a day with a high temperature of \(83^{\circ} \mathrm{F}\). e. Compute the standard deviation of errors. f. Construct a \(99 \%\) confidence interval for \(B\). g. Testing at the \(1 \%\) significance level, can you conclude that \(B\) is different from zero? h. Using \(\alpha=.01\), can you conclude that the correlation coefficient is different from zero?

The following data give information on the ages (in years) and the numbers of breakdowns during the last month for a sample of seven machines at a large company. $$ \begin{array}{l|lllllll} \hline \text { Age (years) } & 12 & 7 & 2 & 8 & 13 & 9 & 4 \\ \hline \text { Number of breakdowns } & 10 & 5 & 1 & 4 & 12 & 7 & 2 \\ \hline \end{array} $$ a. Taking age as an independent variable and number of breakdowns as a dependent variable, what is your hypothesis about the sign of \(B\) in the regression line? (In other words, do you expect \(B\) to be positive or negative?) b. Find the least squares regression line. 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 \(\mathrm{b}\) d. Compute \(r\) and \(r^{2}\) and explain what they mean. e. Compute the standard deviation of errors. f. Construct a \(99 \%\) confidence interval for \(B\). g. Test at the \(2.5 \%\) significance level whether \(B\) is positive. h. At the \(2.5 \%\) significance level, can you conclude that \(\rho\) is positive? Is your conclusion the same as in part g?
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