Understanding Causality and Residuals in Statistics

/ mwriililml UNIT 4 — MILESTONE 4 17118 1 7/1 8 < that's 4% RETAKE ® 17 questions were answered correctly. 1 question was answered incorrectly. T Q@ Which of the following is NOT a guideline for establishing causality? Keep all variables the same to get O O duplicate results. Look for cases where correlation O exists between the variables of a scatterplot. Perform a randomized, controlled O experiment. Take into consideration all the O other possible causes. RATIONALE For causality, the association should be something we observe in slightly varied conditions. So if all variables and conditions are the same, this is not a way to support causality.

/ mwriililml UNIT 4 — MILESTONE 4 17118 —7 CSW@pisning causality Report an issue with this question 2 Q@ Shawna finds a study of American women that had an equation to predict weight (in pounds) from height (in inches): y =-260 + 6.6x. Shawna’s height was 64 inches and her weight was 150 pounds. What is the value of the residual for Shawna's weight and height? 12.4 pounds O -12.4 pounds © O /730 pounds 162.4 pounds O P RATIONALE Recall that to get the residual, we take the actual value - predicted value. So if the actual height of 64 inches and the resulting actual weight is 150 pounds, we simply need the predicted weight. Using the regression line, we can say:. V=-260+6.6(64)=-260+4224=162.4 The predicted weight is 164.4 pounds. So the residual is: residual = actual —predicted =120 1b-1624 b= -124 b CONCEPT — Residuals

/ mwriililml UNIT 4 — MILESTONE 4 17118 3 @ A scientist is comparing parts of the body. The tibia length is plotted against the body length as shown below. Tibia Length Compared to Body Length 72 70 | 1 1 68 66 Body Length (inches) o / 58 6 8 10 12 14 16 18 Tibia Length (inches) Using the best-fit line, approximately how long would the tibia bone be if the body length is 62 inches? 12 inches O

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/ il UNIT 4 — MILESTONE 4 17/18 O 13 inches 11 inches O RATIONALE To get a rough estimate of the length of the tibia when the body length is 62 inches, we go to the value of 62 on the vertical axis and then see where it falls on the best-fit line. This looks to be about 12 inches. CONCEPT — Best-Fit Line and Regression Line Report an issue with this question 4 Q@ This scatterplot shows the performance of a thermocouple using the variables temperature difference and voltage.

Voltage (in mV) N Ul el 4.5 W Ul w N mfmﬂn‘lil;;i:al UNIT 4 — MILESTONE 4 P ol 10 20 30 40 50 60 Temperature Difference (in °C) 70 80 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 linear. 17/18

/ mwriililml UNIT 4 — MILESTONE 4 17118 ~ difference and voltage. Strength: The data pattern is strong. Form: The data pattern is nonlinear. Direction: There is a negative association between temperature O 0O difference and voltage. Strength: The data pattern is weak. Form: The data pattern is nonlinear. Direction: There is a positive O 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 5 @ The scatterplot below shows the relationship between the grams of fat and total calories in different food items.

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/ mwriililml UNIT 4 — MILESTONE 4 17118 525 350 Total Calories 175 0 10 20 30 40 Grams of Fat The equation for the least-squares regression line to this data set is ¥ =13.198x +153.6. What is the predicted number of total calories for a food item that contains 25 grams of fat? O 417.56 © O 483.55 O 383.55 O 549.54 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; ?= 13.198x +153.6 = 13.198(25) + 153.6 = 329.95 + 153.6 = 483.55

/ mwriililml UNIT 4 — MILESTONE 4 17118 — 7 FrediCuoris 1MToOIT oesti-rit Lines Report an issue with this question c @ Which of the following scatterplots shows a correlation affected by inappropriate grouping?

@ :J/ lirfwl‘illilwi:il UNIT 4 — MILESTONE 4 40 35 17/18 30 50 \ i | 45 | | | 40 . o | [ J ‘ ° > 35 ¢ Y | .'. * [ J ‘ [ J 30 | e | 25 | © O * 20 o 15 o * . — [ ] ° .. * ® 10 . - [ J 5 \ | o | | RATIONALE Since this data has 4 distinct groups, by examining overall correlation by all the data, it will not capture how the data in each group is associated. So overall the correlation might not be very strong, but if we look at correlation inside each group, there would be a much stronger relationship. CONCEPT — Cautions about Correlation Report an issue with this question 7 Q@ For the plot below the value of r2 is 0.7783.

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» mfmﬂn‘lil;;i:al UNIT 4 — MILESTONE 4 Cost (in dollars) 750 1500 2250 Distance from MSP (in miles) Which of the following sets of statements is true? About 78% of the variation in distance flown can be explained by the ticket price. The correlation coefficient, r, is 0.606. About 22% of the variation in ticket price can be explained by the distance flown. The correlation coefficient, r, is 0.606. About 78% of the variation in ticket price can be explained by the distance flown. The correlation coefficient, r, is 0.882. 3000 17/18

/ mwriililml UNIT 4 — MILESTONE 4 17118 MIILT LAll YT TANMIAITNITU VY LIT distance flown. O The correlation coefficient, r, is 0.882. 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, . We can also note that r = 'Jf_zzv’ 0.778 =0.882, CONCEPT — Coefficient of Determination/r*2 Report an issue with this question & @ 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 al 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

D yiillii UNIT 4 — MILESTONE 4 v eSS ) O 0.92 O 0.85 2O 1.00 O 0.71 RATIONALE In order to get the correlation, we can use the formula: x—x)(u)=(2—3.6)(6— 9.2, ,6-36, 14-9.2 1-3.6,,4-9.2 & Ny )= Mgy A T 2.1 4.1 2.1 4.1 n comelation = — — 2. ( n-—1 i=1 )=1 Correlation can be quickly calculated by using Excel. Enter the values and use the function "=CORREL(". A B C 1 | Grade 'Reading Level 2 | 2 6 3 6 14 4 5 12 5 | 4 10 6 | 1 4 7 8 |=CORREL(B2:B6,C2:C6) A B C 1 :Grade ‘Reading Level 2 | 2 6 3 | 6 14 4 5 12 5 | 4 10 6 1| 4 7 8 1

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» mfmﬂn‘lil;;i:al UNIT 4 — MILESTONE 4 —7 Lorlreidtuori Report an issue with this question ° @ 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. Cost of Airplane Ticket vs Miles Flown 700 L J __ 525 N C c 350 o oy S 8 ® 175 e O o) o) 750 1500 2250 3000 Distance from MSP (in miles) Which answer choice correctly indicates the explanatory and response variables for the scatterplot? Explanatory variable: Cost Response variable: Distance Explanatory variable: Miles flown Response variable: Minneapolis 17/18

mfmﬂn‘lil;;i:al UNIT 4 — MILESTONE 4 Response variable: Miles flown Explanatory variable: Distance Response variable: Cost 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 10 @ Jesse takes two data points from the weight and feed cost data set to calculate a slope, or average rate of change. A rat weighs 3.5 pounds and costs $4.50 per week to feed, while a Beagle weighs 30 pounds and costs $9.20 per week to feed. Using weight as the explanatory variable, what is the slope of the line between these two points? Answer choices are rounded to the nearest hundredth. O $1.60 / Ib. © O $018/ Ib. 17/18

mfmﬂn‘lil;;i:al UNIT 4 — MILESTONE 4 $0.31/ Ib. O RATIONALE | V2= In order to get slope, we can use the formula: slope = NEIR Using the information provided, the two points are: (3.5 Ib., $4.50) and (30 Ib., $9.20). We can note that: y2-y1 _ $9.20-$4.50 _ $4.70 = =$0.18 /Ib x2—x1 30b-3.51b 26.21b ’ slope = CONCEPT — Linear Equation Algebra Review Report an issue with this question 1 @ A correlation coefficient between average temperature and ice cream sales is most likely to be between —1 and -2 O between O and —1 O between O and 1 © O between 1 and 2 O RATIONALE 17/18

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/ mwriililml UNIT 4 — MILESTONE 4 17118 CONCEPT — Positive and Negative Correlations Report an issue with this question 2 @ Which statement about correlation is FALSE? Correlation between the variables O of the data set can be measured. Correlation is used to define the variables of only non-linearly © O related data sets. Correlation is the degree to which O the two variables of a data set resemble each other. The correlation of a data set can be O positive, negative, or O. RATIONALE Recall that correlation is used for linear association between 2 quantitative variables, NOT for non-linearly related variables. CONCEPT — Correlation Report an issue with this question

% il UNIT 4 1S ' — MILESTONE 4 Which of the following scatterplots shows an outlier in the y- direction? Test Scores Test Scores Test Scores Number of Hours of Sleep vs. Test Results 100 ‘ ‘ ‘ o 000 o 1 2 3 4 5 6 7 8 9 10 Number of Hours of Sleep Number of Hours of Sleep vs. Test Results 100 | | | % | e e o 1 - 3 4 5 6 7 8 9 10 Number of Hours of Sleep Number of Hours of Sleep vs. Test Results 100 | | | | % | e e o 1 - 3 4 5 6 7 8 9 10 Number of Hours of Sleep 17/18

/ mwriililml UNIT 4 — MILESTONE 4 17118 o w 10| ' ' ‘ 1 % 60 e fil_" 50 O 8 40 30 20 10 ; 0 I o 1 - 3 4 5 6 7 8 9 10 Number of Hours of Sleep RATIONALE To have an outlier in the y-direction the outlier must be in the range of X data but outside the range of y-data. This outlier is outside of the data in the y direction, lying below all of the data. CONCEPT — Outliers and Influential Points Report an issue with this question 14‘ 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 y =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 O 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.

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/ mwriililml UNIT 4 — MILESTONE 4 17118 Q V.0 YUULIUO, VII avTiliaycT, vWIICII LT baby's age increases by 1 month. The weight of a baby decreases by 0.8 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 5 @ Which of the following statements is TRUE? Only a correlation equal to O O implies causation. A high correlation is insufficient to ® O establish causation on its own. A correlation of 1 or -1 implies O causation.

% il UNIT 4 — MILESTONE 4 17/18 al < DLIUIIny ITIALTU, LHHO CUILIUILIVILI O implies causation between the two variables. 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. Additional information is required to know something is causal, like seeing the association validated in an experimental design. CONCEPT — Correlation and Causation Report an issue with this question 6 @ 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. |dentify the lurking variable that is causing an increase in both the number of cars owned and the average number of citizens with health insurance. O Average mileage per vehicle Average income per household

/ mwriililml UNIT 4 — MILESTONE 4 17118 The number of cars on the road O The number of citizens in the (O 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 17 @ For a set of data, x is the explanatory variable. Ilts mean is 8.2, and its standard deviation is 1.92. For the same set of data, y is the response variable. Its mean is 13.8, and its standard deviation is 3.03. The correlation was found to be 0.223. Select the correct slope and y-intercept for the least-squares line. Answer choices are rounded to the hundredths place. Slope = 0.35 y-intercept = 3.37

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/ mwriililml UNIT 4 — MILESTONE 4 17118 © y-intercept = 10.93 Slope =0.14 y-intercept =12.65 Slope =0.14 y-intercept = 6.27 RATIONALE We first want to get the slope. We can use the formula: 5, (3.03) lope =by =r-| -£ |=0.223-[ 2221 =0.35 >1ope = b1 r[sx] 1.92 To then get the intercept, we can solve for the y-intercept by using the following formula: We know the slope, £1, and we can use the mean of x and the mean of y for the variables % and i to solve for the y-intercept, Dy, ¥V =bg+byx 13.8 =by+0.35(8.2) 13.8=by+2.87 10.93=by CONCEPT — Finding the Least-Squares Line Report an issue with this question 18

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/ mwriililml UNIT 4 — MILESTONE 4 17118 she is enrolled in. The information is shown in the table below. Study Table Study-Hours S + ‘7 o 0 2 4 6 8 10 12 Number of Semester Credits If Rhonda is taking four credits for the fall semester, how many hours per week will she study?

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/| mwriililml UNIT 4 — MILESTONE 4 17/18 RATIONALE If we use the scatterplot and note the value above 4 credit hours on the horizontal axis, we find this value is also 4 on the vertical axis. She should expect to study 4 hours. CONCEPT — Scatterplot Report an issue with this question % About Contact Us Privacy Policy Terms of Use © 2023 SOPHIA Learning, LLC. SOPHIA is a registered trademark of SOPHIA Learning, LLC.

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INTRO TO STATISTICS MILESTONE 4

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