Understanding Correlation and Scatterplots in Data Analysis

14/18 that's 78% RETAKE 14 questions were answered correctly . 4 questions were answered incorrectly . 1 A correlation coefficient between number of miles driven and number of gallons of gas remaining is most likely to be __________.  between 0 and -1  between 0 and 1  between -1 and -2  between 1 and 2 RATIONALE As the number of miles increases more gas is required, so there should be less gas in the tank. This would be a negative relationship and we would expect the value to be between 0 and -1. CONCEPT Positive and Negative Correlations Report an issue with this question 2 This scatterplot shows the performance of a thermocouple using the variables temperature difference and voltage. Select the answer choice that accurately describes the data's form, direction, and strength in the scatterplot.  Form: The data pattern is linear. Direction: There is a positive association between temperature difference and voltage.

Strength: The data pattern is strong.  Form: The data pattern is nonlinear. Direction: There is a positive association between temperature difference and voltage. Strength: The data pattern is weak.  Form: The data pattern is linear. Direction: There is a negative association between temperature difference and voltage. Strength: The data pattern is strong.  Form: The data pattern is nonlinear. Direction: There is a negative association between temperature difference and voltage. Strength: The data pattern is weak. RATIONALE If we look at the data, it looks as if a straight line captures the relationship, so the form is linear. The slope of the line is positive, so it is increasing. Finally, since the dots are closely huddled around each other in a linear fashion, it looks strong. CONCEPT Describing Scatterplots Report an issue with this question 3 A basketball player recorded the number of baskets he could make depending on how far away he stood from the basketball net. The distance from the net (in feet) is plotted against the number of baskets made as shown below.

Using the best-fit line, approximately how many baskets can the player make if he is standing ten feet from the net?  9 baskets  8 baskets  5 baskets  3 baskets RATIONALE To get a rough estimate of the number of baskets made when standing 10 feet from the net, we go to the value of 10 on the horizontal axis and then see where it falls on the best-fit line. This looks to be about 5 baskets. CONCEPT Best-Fit Line and Regression Line Report an issue with this question 4 Which statement about correlation is FALSE?  Correlation is a quantitative measure of the strength of a linear association between two variables.  Correlation is a quantitative measure of the strength of a non-linear association between two variables.  A correlation of -1 or 1 corresponds to a perfectly linear relationship.  Correlation is measured by r, the correlation coefficient which has a value between -1 and 1. RATIONALE Correlation specifically measures the strength and direction of a linear association between two variables. So we can't use it for all variables or for non-linear associations. CONCEPT

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Correlation Report an issue with this question 5 Shawna reads a scatterplot that displays the relationship between the number of cars owned per household and the average number of citizens who have health insurance in neighborhoods across the country. The plot shows a strong positive correlation. Shawna recalls that correlation does not imply causation. In this example, Shawna sees that increasing the number of cars per household would not cause members of her community to purchase health insurance. Identify the lurking variable that is causing an increase in both the number of cars owned and the average number of citizens with health insurance.  Average income per household  Average mileage per vehicle  The number of cars on the road  The number of citizens in the United States RATIONALE Recall that a lurking variable is something that must be related to the outcome and explanatory variable that when considered can help explain a relationship between 2 variables. Since higher income is positively related to owning more cars and having health insurance, this variable would help explain why we see this association. CONCEPT Correlation and Causation Report an issue with this question 6 Which of the following scatterplots best shows an outlier in ONLY the x- direction?

    RATIONALE To have an outlier in the x-direction the outlier must be in the range of y data but outside the range of x-data. CONCEPT Outliers and Influential Points Report an issue with this question 7 Data for weight (in pounds) and age (in months) of babies is entered into a statistics software package and results in a regression equation of ŷ = 17 + 0.8x. What is the correct interpretation of the slope if the weight is the response variable and the age is the explanatory variable?  The weight of a baby decreases by 0.8 pounds, on average, when the baby's age increases by 1 month.  The weight of a baby increases by 0.8 pounds, on average, when the baby's age increases by 1 month.  The weight of a baby decreases by 17 pounds, on average, when the baby's age increases by 1 month.  The weight of a baby increases by 17 pounds, on average, when the baby's age increases by 1 month. RATIONALE When interpreting the linear slope we generally substitute in a value of 1. So we can note that, in general, as x increases by 1 unit the slope tells us how the outcome changes. So for this equation we can note as

x (age) increases by 1 month, the outcome (weight) will increase by 0.8 pounds on average. CONCEPT Interpreting Intercept and Slope Report an issue with this question 8 Which of the following statements is TRUE?  Only a correlation of 1 implies causation.  Only a correlation of 0 implies causation.  High correlation always implies causation.  High correlation does not always imply causation. RATIONALE Recall that correlation doesn't imply causation. Causation is a direct change in one variable causing a change in some outcome. Correlation is simply a measure of association. It is required for causation, but alone does not mean something is causal. CONCEPT Correlation and Causation Report an issue with this question 9 Shawna finds a study of American women that has an equation to predict weight (in pounds) from height (in inches): ŷ = -260 + 6.6x. Shawna's mom’s height is 68 inches and her weight is 179 pounds. What is the residual of weight and height for Shawna's mom?  921.4 pounds  -9.8 pounds  9.8 pounds  188.8 pounds

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RATIONALE Recall that to get the residual, we take the actual value - predicted value. So if the actual height of 68 inches and the resulting actual weight is 179 pounds, we simply need the predicted weight. Using the regression line, we can say: The predicted weight is 188.8 pounds. So the residual is: CONCEPT Residuals Report an issue with this question 10 The table below shows the grade and reading level for 5 students. Grade Reading Level Student 1 2 6 Student 2 6 14 Student 3 5 12 Student 4 4 10 Student 5 1 4 For grade, the mean is 3.6 and the standard deviation is 2.1. For reading level, the mean is 9.2 and the standard deviation is 4.1. Using the formula below or Excel, find the correlation coefficient, r, for this set of students. Answer choices are rounded to the nearest hundredth.  0.71

 0.92  0.85  1.00 RATIONALE In order to get the correlation, we can use the formula: Correlation can be quickly calculated by using Excel. Enter the values and use the function "=CORREL(". CONCEPT Correlation Report an issue with this question 11 Which of the following scatterplots shows a correlation affected by non- linearity?     RATIONALE Recall that correlation is for linear association, so if a graph shows curvature, then it would not be adequately captured by using correlation. This plot clearly shows a non-linear graph. CONCEPT

Cautions about Correlation Report an issue with this question 12 A bank manager declares, with help of a scatterplot, that the number of health insurances sold may have some association with the number of inches it snows. How many policies were sold when it snowed 2 to 4 inches?  240  470  210  350 RATIONALE In order to find the total number of policies between 2 and 4 inches, we must add the three values of 10 in that interval. At 2 inches, there were 100 policies. At 3 inches, there were 110 policies. At 4 inches, there were 140 policies. So the total is 100 + 110 + 140 = 350 policies. CONCEPT Scatterplot Report an issue with this question 13 Raoul lives in Minneapolis and he is planning his spring break trip. He creates the scatterplot below to assess how much his trip will cost.

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Which answer choice correctly indicates the explanatory and response variables for the scatterplot?  Explanatory variable: Miles flown Response variable: Minneapolis  Explanatory variable: Distance Response variable: Cost  Explanatory variable: Minneapolis Response variable: Miles flown  Explanatory variable: Cost Response variable: Distance RATIONALE The explanatory variable is what is along the horizontal axis, which is distance. The response variable is along the vertical axis, which is cost. CONCEPT Explanatory and Response Variables Report an issue with this question 14 The scatterplot below shows the relationship between the grams of fat and total calories in different food items. The equation for the least-squares regression line to this data set is . What is the predicted number of total calories for a food item that contains 25 grams of fat?

 549.54  383.55  483.55  417.56 RATIONALE In order to get the predicted calories when the grams of fat is equal to 25, we simply substitute the value 25 in our equation for x. So we can note that: CONCEPT Predictions from Best-Fit Lines Report an issue with this question 15 For the plot below the value of r 2 is 0.7783. Which of the following sets of statements is true?  About 78% of the variation in ticket price can be explained by the distance flown. The correlation coefficient, r, is 0.882.  About 22% of the variation in ticket price can be explained by the distance flown. The correlation coefficient, r, is 0.606.  About 88% of the variation in ticket price can be explained by the distance flown. The correlation coefficient, r, is 0.882.  About 78% of the variation in distance flown can be explained by the ticket price.

The correlation coefficient, r, is 0.606. RATIONALE The coefficient of determination measures the percent of variation in the outcome, y, explained by the regression. So a value of 0.7783 tells us the regression with distance, x, can explain about 77.8% of the variation in cost, y. We can also note that r = . CONCEPT Coefficient of Determination/r^2 Report an issue with this question 16 Which of the following is a guideline for establishing causality?  Look for cases where correlation does not exist between the variables.  The experiment performed should be controlled and randomized.  Do not consider other possible causes.  Check if the effect is absent when the response variable is present. RATIONALE Finding an association inside an experimental design controls for many other outside influences and helps to ensure that an explanatory variable always precedes the response. This helps to support, with strong confidence, that the association is causal. CONCEPT Establishing Causality Report an issue with this question 17 Jesse takes two data points from the weight and feed cost data set to calculate a slope, or average rate of change. A hamster weighs half a pound and costs $2 per week to feed, while a Labrador Retriever weighs 62.5 pounds and costs $10 per week to feed.

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Using weight as the explanatory variable, what is the slope of a line between these two points? Answer choices are rounded to the nearest hundredth.  $7.75 / lb.  $6.25 / lb.  $4.00 / lb.  $0.13 / lb. RATIONALE In order to get slope, we can use the formula: . Using the information provided, the two points are: (0.5 lb., $2) and (62.5 lb., $10). We can note that: CONCEPT Linear Equation Algebra Review Report an issue with this question 18 For ten students, a teacher records the following scores of two assessments, Quiz 1 and Test. Quiz 1 (x) Test (y) 15 20 12 15 10 12 14 18 10 10

8 13 6 12 15 10 16 18 13 15 Mean 11.9 14.3 Standard Deviation 3.3 3.5 The correlation of Quiz 1 and Test is 0.568. Given the information below, what is the slope and y-intercept for the least-squares line of the Quiz 1 scores and Test scores? Answer choices are rounded to the hundredths place.  Slope = 0.54 y-intercept = 7.87  Slope = 0.54 y-intercept = 4.18  Slope = 0.60 y-intercept = 7.16  Slope = 0.60 y-intercept = 3.32 RATIONALE We first want to get the slope. We can use the formula: To then get the intercept, we can solve for the y-intercept by using the following formula:

We know the slope, , and we can use the mean of x and the mean of y for the variables and to solve for the y-intercept, . CONCEPT Finding the Least-Squares Line Report an issue with this question  About  Contact Us  Privacy Policy  Cookie Policy  Terms of Use  Your Privacy Choices © 2023 SOPHIA Learning, LLC. SOPHIA is a regis

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