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Chapter 13: Problem 53

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

Short Answer

Expert verified
a. Positive correlation b. Positive correlation c. Positive correlation d. Negative correlation e. Zero correlation.

Step by step solution

01

Example 1: SAT scores and GPAs of students

Generally, students with higher SAT scores also tend to have higher GPAs as both indicators relate to academic performance. Therefore, there should be a positive correlation.
02

Example 2: Stress level and blood pressure of individuals

Increased stress levels often cause an increase in blood pressure. Thus, a positive correlation is expected here.
03

Example 3: Amount of fertilizer used and yield of corn per acre

Increased amounts of fertilizers tend to increase the yield of corn per acre. Therefore, there should be a positive correlation.
04

Example 4: Ages and prices of houses

Older houses tend to be cheaper than new ones because they may require more maintenance and have outdated features. So, a negative correlation is expected.
05

Example 5: Heights of husbands and incomes of their wives

The height of a husband has no direct impact on his wife's income. Therefore, a zero correlation is expected here.

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Key Concepts

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

SAT scores and GPA
SAT scores and GPA are closely related as both are measures of academic success. Students who perform well on the SAT often have strong study habits and critical thinking skills that also lead to higher GPAs. These two metrics give a snapshot of a student's academic ability:
  • SAT scores are standardized tests assessing mathematical, verbal, and writing skills.
  • GPA measures academic achievement based on grades over time in coursework.

The relationship between SAT scores and GPA is generally positive, meaning that as one increases, the other tends to increase as well. This is because both reflect similar types of academic performance and proficiency in educational contexts. Understanding this can help students focus on aligning their study habits and test preparation for both areas.
Stress Level and Blood Pressure
Stress level and blood pressure have a significant correlation, primarily because stress triggers physiological responses in the body. Stress activates the body's fight-or-flight response, leading to elevated blood pressure:
  • Chronic stress results in persistent high blood pressure, which is a risk factor for heart disease.
  • Temporary stress can cause short-term spikes in blood pressure, even if a person is generally healthy.

The positive correlation between stress and blood pressure means that as stress levels rise, blood pressure is likely to increase too. For many individuals, recognizing this connection can be a step toward better health management, encouraging stress reduction techniques like mindfulness and exercise to maintain healthier blood pressure levels.
Fertilizer and Crop Yield
The relationship between the amount of fertilizer used and the yield of crops such as corn is straightforward: more fertilizer often results in a higher crop yield. Fertilizers provide essential nutrients that plants need for growth:
  • Nitrogen, phosphorus, and potassium are common nutrients found in fertilizers that promote plant health and productivity.
  • Proper use of fertilizers can significantly enhance agricultural outputs.

This relationship shows a positive correlation but with diminishing returns at higher levels of fertilizer application, where too much can lead to negative environmental impacts. Farmers are encouraged to balance fertilizer use for optimal yield, considering both economic benefits and environmental stewardship.
Housing Market Trends
A key trend in the housing market is the relationship between the age and price of houses. Typically, older homes are priced lower because they may require more repairs or lack modern features desired by buyers:
  • Older homes might offer solid construction but could also need updates in wiring, plumbing, or aesthetic appeal.
  • Newer homes often feature modern amenities and design trends, which attract higher prices.

Thus, there is usually a negative correlation between the age of a house and its price. Buyers and investors should consider both cost and the potential need for renovations when evaluating older properties.
Heights of Husbands and Incomes of Their Wives
The heights of husbands and the incomes of their wives typically show zero correlation, meaning no predicted connection exists between these two variables. This lack of correlation arises because the attributes do not influence one another:
  • Height is a physical trait determined by genetics.
  • Income often depends on factors like education, career choice, and professional experience.

Recognizing this can help individuals avoid assumptions based on unrelated characteristics, focusing attention on more meaningful attributes when assessing both physical and economic outcomes.

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Most popular questions from this chapter

The following table gives information on GPAs and starting salaries (rounded to the nearest thousand dollars) of seven recent college graduates. $$ \begin{array}{l|rrrrrrr} \hline \text { GPA } & 2.90 & 3.81 & 3.20 & 2.42 & 3.94 & 2.05 & 2.25 \\ \hline \text { Starting salary } & 48 & 53 & 50 & 37 & 65 & 32 & 37 \\ \hline \end{array} $$ a. With GPA as an independent variable and starting salary as a dependent variable, compute \(\mathrm{SS}_{x x}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x y}\) b. Find the least squares regression line. c. Interpret the meaning of the values of \(a\) and \(b\) calculated in part b. d. Calculate \(r\) and \(r^{2}\) and briefly explain what they mean. e. Compute the standard deviation of errors. fonstruct a \(95 \%\) confidence interval for \(B\). g. Test at a \(1 \%\) significance level whether \(B\) is different from zero. h. Test at a \(1 \%\) significance level whether \(\rho\) is positive.

The following table gives the 2015 total payroll (in millions of dollars) and the percentage of games won during the 2015 season by each of the National League baseball teams. $$ \begin{array}{lcc} \hline \text { Team } & \begin{array}{c} \text { Total Payroll } \\ \text { (millions of dollars) } \end{array} & \begin{array}{c} \text { Percentage of } \\ \text { Games Won } \end{array} \\ \hline \text { Arizona Diamondbacks } & 92 & 49 \\ \text { Atlanta Braves } & 98 & 41 \\ \text { Chicago Cubs } & 119 & 60 \\ \text { Cincinnati Reds } & 117 & 40 \\ \text { Colorado Rockies } & 102 & 42 \\ \text { Los Angeles Dodgers } & 273 & 57 \\ \text { Miami Marlins } & 68 & 44 \\ \text { Milwaukee Brewers } & 105 & 42 \\ \text { New York Mets } & 101 & 56 \\ \text { Philadelphia Phillies } & 136 & 39 \\ \text { Pittsburgh Pirates } & 88 & 61 \\ \text { San Diego Padres } & 101 & 46 \\ \text { San Francisco Giants } & 173 & 52 \\ \text { St. Louis Cardinals } & 121 & 62 \\ \text { Washington Nationals } & 165 & 51 \\ \hline \end{array} $$ a. Find the least squares regression line with total payroll as the independent variable and percentage of games won 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 percentage of games won by a team with a total payroll of \(\$ 150\) million.

The following table gives information on the amount of sugar (in grams) and the calorie count in one serving of a sample of 13 different varieties of cereal. $$ \begin{array}{l|rrrrrrr} \hline \text { Sugar (grams) } & 4 & 15 & 12 & 11 & 8 & 6 & 7 \\ \hline \text { Calories } & 120 & 200 & 140 & 110 & 120 & 80 & 190 \\ \hline \text { Sugar (grams) } & 2 & 7 & 14 & 20 & 3 & 13 & \\ \hline \text { Calories } & 100 & 120 & 190 & 190 & 110 & 120 & \\ \hline \end{array} $$ a. Construct a scatter diagram for these data. Does the scatter diagram exhibit a linear relationship between the amount of sugar and the number of calories per serving? b. Find the regression equation of the number of calories on the amount of sugar. c. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part b. d. Plot the 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. Calculate the calorie count for a cereal with 16 grams of sugar per serving. f. Estimate the calorie count for a cereal with 52 grams of sugar per serving. Comment on this finding.

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. Here price is the dependent variable. $$ \begin{array}{l|rrrr} \hline \text { Percentage containing ads } & 37 & 43 & 58 & 49 \\ \hline \text { Price (\$) } & 5.50 & 6.95 & 4.95 & 5.75 \\ \hline \text { Percentage containing ads } & 70 & 28 & 65 & 32 \\ \hline \text { Price (\$) } & 3.95 & 8.25 & 5.50 & 6.75 \\ \hline \end{array} $$ a. Find the standard deviation of errors. b. Compute the coefficient of determination. What percentage of the variation in price is explained by the least squares regression of price on the percentage of magazine space containing ads? What percentage of this variation is not explained?

Will you expect a positive, zero, or negative linear correlation between the two variables for each of the following examples? a. Grade of a student and hours spent studying b. Incomes and entertainment expenditures of households c. Ages of women and makeup expenses per month d. Price of a computer and consumption of Coca-Cola e. Price and consumption of wine
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