Numerical Associations - Extra Practice
Practice Questions
Exercise 1
An analyst calculates the correlation coefficient between \(X\) = the height of a plant (in centimeters) and \(Y\) = its water consumption (in milliliters), to be \(r = 0.75\). If the analyst converts the height measurements to inches (\(X_2 = X \timex 0.3937\)) and the water consumption to fluid ounces (\(Y_2 = Y * 0.03381\)), what will the correlation coefficient be between \(X_2\) and \(Y_2\)?
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Answer: \(r = 0.75\). Changing the units of measurement does not change the correlation coefficient.
Exercise 2
A researcher studies the relationship between student test scores (\(X\)) and the number of hours spent sleeping (\(Y\)). They find a correlation coefficient of \(r = 0.58\). What would the correlation coefficient be if they swapped the variables so that sleeping hours is \(X\) and test scores is \(Y\)?
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Answer: \(r = 0.58\). Switching the explanatory (\(X\)) and response (\(Y\)) variables does not affect the value of \(r\).
Exercise 3
A study on children’s development finds that vocabulary size and shoe size are strongly positively correlated ((r = 0.82)). A local news headline claims, “Buying larger shoes causes your child to learn more words.” Why is this conclusion flawed?
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Answer: Correlation does not imply causation. A high \(r\) value shows a linear pattern but does not prove that changing one variable directly changes the other.
Exercise 4
A dataset plotting a company’s monthly production volume and its total factory overhead costs shows a perfect U-shaped quadratic curve. Every single data point sits exactly on this curved line. What is the most likely value of the Pearson correlation coefficient (\(r\)) for this dataset?
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Answer: Close to 0. The correlation coefficient only measures the strength and direction of linear relationships and is blind to curved patterns.
Exercise 5
A statistics student inputs a dataset into a software program to find the relationship between car weights and fuel economy. The software outputs a correlation coefficient of \(r = -1.14\). How should the student interpret this result?
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Answer: A calculation error occurred. The value of \(r\) must strictly fall within the range of -1 to +1.
Exercise 6
A textbook lists the correlation coefficient between outdoor temperature and daily hot chocolate sales at a cafe as \(r = -0.72\). What unit of measurement accompanies this \(r\) value?
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Answer: It has no units. The correlation coefficient is a standardized, pure dimensionless number.
Exercise 7
A real estate analyst studies the relationship between house size in square feet (\(X\)) and selling price in thousands of dollars (\(Y\)). The data yields a mean house size of \(\bar{x} = 1,800\) with a standard deviation of \(S_x = 400\). The mean selling price is \(\bar{y} = 350\) with a standard deviation of \(S_y = 120\). The correlation coefficient between the two variables is \(r = 0.85\). Calculate the slope (\(b_1\)) and the y-intercept (\(b_0\)) of the least-squares regression line.
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Answer: Slope \(b_1 = 0.255\), Y-intercept \(b_0 = -109\). The equation of the regression line is \(\hat{y} = -109 + 0.255 x\)
Exercise 8
Assume that the least squares regression to predict \(y\) = house prices (in thousands of dollars) in terms of their \(x\) = size (in square feet) is \(\hat{y} = -109 + 0.255x\). Provide a verbal interpretation for the slope (\(b_1 = 0.255\)) in the context of house sizes and selling prices.
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Answer: For each additional square foot of house size, the predicted selling price increases by 0.255 thousand dollars (or $255). The slope represents the average change in the predicted response for a one-unit increase in the explanatory variable.
Exercise 9
Refer to the previous least squares regression model to predict \(y\) = house prices (in thousands of dollars) in terms of their \(x\) = size (in square feet) igiven by \(\hat{y} = -109 + 0.255x\). Provide a verbal interpretation for the intercept (\(b_0 = -109\)) in the context of house sizes and selling prices.
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Answer: The predicted selling price for a house with 0 square feet is -109 thousand dollars, which makes no practical sense. The y-intercept represents the predicted value of \(y\) when \(x = 0\).
Exercise 10
A high school counselor tracks the relationship between student attendance rates in percentages (\(x\)) and final exam grades (\(y\)). They obtained a simple linear model to preidct \(y\) given \(x\): \[\hat{y} = -30 + 114 = 84\]
A specific student in the dataset had an attendance rate of \(95\%\) and actually scored a \(79\) on the final exam. Calculate the residual for this student and interpret what the sign of the residual means.
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Answer: Actual \(y = 79\). Predicted \(\hat{y} = 84\). Residual \(\hat{e} = 79 - 84 = -5\). The negative sign means the student scored lower than the model predicted.