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Chapter 13: Problem 52

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

Short Answer

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

Step by step solution

01

Correlation between Grade and Study Hours

Consider the relationship between the grades of a student and the hours spent studying. As the hours spent studying increase, it would be reasonable to expect a student's grades to improve due to increased preparation. Therefore, the correlation between these two variables is expected to be positive.
02

Correlation between Incomes and Entertainment Expenditures

Consider the relationship between incomes and entertainment expenditures. As income increases, households would typically spend more on entertainment. Therefore, a positive correlation is expected between these two variables.
03

Correlation between Age and Makeup Expenses

If we consider the relationship between the age of women and their monthly expenditure on makeup, it can be assumed that as women grow older, their expenditure on makeup might decrease. As such, one might expect a negative correlation between these two variables.
04

Correlation between Computer Price and Consumption of Coca-Cola

Consider the relationship between the price of a computer and the consumption of Coca-Cola. These two variables don't seem to have a logical or plausible direct relationship. Therefore, the expected correlation would be close to zero, indicating no linear correlation.
05

Correlation between Price and Consumption of Wine

Looking into the relationship between the price and consumption of wine, one could assume that when the price goes up, the consumption might go down due to the increased cost. Therefore, a negative correlation would be expected between these two variables.

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Key Concepts

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

Linear Correlation
Linear correlation is a measure that describes the strength and direction of a relationship between two variables. It helps us understand if changes in one variable can predict changes in another. When assessing linear correlation:
  • A correlation coefficient, often represented by the symbol \( r \), ranges from -1 to 1.
  • A perfect positive correlation would have \( r = 1 \), while a perfect negative correlation would have \( r = -1 \).
  • An \( r \) value around 0 suggests no linear relationship.
Linear correlation is a foundational element in statistics, often used to make predictions or to determine the strength of a relationship between two continuous variables. Understanding this concept is crucial when analyzing data sets and patterns in various fields such as economics, psychology, and social sciences.
Positive Correlation
Positive correlation occurs when two variables move in the same direction. This means that as one variable increases, the other also increases, and vice versa. For example, the relationship between the hours a student studies and their grades is typically positive. As study hours increase, grades tend to improve due to better preparation.
  • Positive correlation does not imply causation, it simply indicates a relationship.
  • The closer the correlation coefficient \( r \) is to 1, the stronger the positive linear relationship.
In applied scenarios like household incomes and entertainment expenditures, a rise in income is usually associated with increased spending on leisure, showcasing a positive correlation. It's essential to visualize data with scatter plots to see this upward trend.
Negative Correlation
Negative correlation is observed when two variables move in opposite directions. In this context, as one variable increases, the other decreases, and vice versa.
  • The correlation coefficient \( r \) will be between 0 and -1 for negative correlations.
  • A correlation of \( r = -1 \) suggests a perfect negative linear relationship.
For instance, as the price of wine increases, we might notice a decrease in its consumption. This creates an inverse relationship due to possibly reduced affordability for consumers. Similarly, as women age, expenditures on makeup might decrease, suggesting a negative correlation. Negative correlations should be explored using visualization tools to better understand these inverse relationships.
Variables Relationship
The relationship between variables is of paramount importance in statistics, as it allows for better understanding and prediction of data patterns. A clear understanding of whether these relationships are positive, negative, or non-existent helps in the application of appropriate statistical methods.
  • Variables with no correlation are independent of each other. Such relations often have a correlation coefficient \( r \) near zero.
  • Analyzing relationships often involves considering external factors or potential additional variables that might influence the behavior of the analyzed variables.
For example, considering the price of a computer and Coca-Cola consumption, there's likely no meaningful relationship, indicating these variables are unrelated. Analyzing relationships, whether linear or non-linear, aids in forming hypotheses and testing theories in scientific research and practical problem-solving scenarios.

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Most popular questions from this chapter

Consider the formulas for calculating a prediction interval for a new (specific) value of \(y\). For each of the changes mentioned in parts a through \(\mathrm{c}\) below, state the effect on the width of the confidence interval (increase, decrease, or no change) and why it happens. Note that besides the change mentioned in each part, everything else such as the values of \(a, b, \bar{x}, s_{e}\), and \(S S_{x x}\) remains unchanged. a. The confidence level is increased. b. The sample size is increased. c. The value of \(x_{0}\) is moved farther away from the value of \(\bar{x}\). d. What will the value of the margin of error be if \(x_{0}\) equals \(\bar{x}\)?

Explain the difference between a simple and a multiple regression model.

Explain the meaning of coefficient of determination.

The following table gives information on the incomes (in thousands of dollars) and charitable contributions (in hundreds of dollars) for the last year for a random sample of 10 households. $$ \begin{array}{cc} \hline \text { Income } & \text { Charitable Contributions } \\ \hline 76 & 15 \\ 57 & 4 \\ 140 & 42 \\ 97 & 33 \\ 75 & 5 \\ 107 & 32 \\ 65 & 10 \\ 77 & 18 \\ 102 & 28 \\ 53 & 4 \\ \hline \end{array} $$ a. With income as an independent variable and charitable contributions as a dependent variable, compute \(\mathrm{SS}_{x}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x y}\) b. Find the regression of charitable contributions on income. c. Briefly explain the meaning of the values of \(a\) and \(b\). d. Calculate \(r\) and \(r^{2}\) and briefly explain what they mean. e. Compute the standard deviation of errors. f. Construct a 99 \% confidence interval for \(B\). g. Test at a \(1 \%\) significance level whether \(B\) is positive. h. Using a \(1 \%\) significance level, can you conclude that the linear correlation coefficient is different from zero?

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. $$ \begin{array}{lcc} \hline \text { Bagel } & \text { Calories } & \text { Fat (grams) } \\ \hline \text { Asiago Cheese } & 330 & 6.0 \\ \text { Blueberry } & 340 & 1.5 \\ \text { Cinnamon Crunch } & 420 & 6.0 \\ \text { Cinnamon Swirl \& Raisin } & 320 & 2.0 \\ \text { Everything } & 300 & 2.5 \\ \text { French Toast } & 350 & 4.0 \\ \text { Plain } & 290 & 1.5 \\ \text { Sesame } & 310 & 3.0 \\ \hline \end{array} $$ With calories as the dependent variable and fat content as the independent variable, find the following: a. \(\mathrm{SS}_{x x}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x y}\) b. Standard deviation of errors c. SST, SSE, and SSR d. Coefficient of determination
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