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The scores of the quizzes of five students in both English and Science are:
English Science
Student 1 6 8
Student 2 5 5
Student 3 9 6
Student 4 4 7
Student 5 8 9
For English, the mean is 6.4 and the standard deviation is 2.0. For Science, the mean is 7 and the standard deviation is 1.6.
Using the formula below or Excel, find the correlation coefficient, r, for this set of scores. Answer choices are rounded to the nearest hundredth.
0.50
0.23
0.05
0.42
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 2
This scatterplot shows the number of hours a student slept every night and his or her grade point average.
The equation for the least-squares regression line to this data is: ŷ = 0.375x + 1.33.
What is the predicted GPA for a student who sleeps 2.5 hours per day? Answer choices are rounded to the hundredths place.
2.64
2.46
2.08
2.27
RATIONALE
In order to get the predicted GPA when the hours of sleep is equal to 2.5, we simply substitute the value 2.5 in our equation for x. So we can note that:
CONCEPT
Predictions from Best-Fit Lines 3
Data for price and thickness of soap is entered into a statistics software package and results in a regression equation of ŷ = 0.4 + 0.2x.
What is the correct interpretation of the slope if the price is the response variable and the thickness is an explanatory variable?
The price of the soap decreases by $0.20, on average, when the thickness increases by 1 cm.
The price of the soap increases by $0.40, on average, when the thickness increases by 1 cm.
The price of the soap decreases by $0.40, on average, when the thickness increases by 1 cm.
The price of the soap increases by $0.20, on average, when the thickness increases by 1 cm.
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 (thickness) increases by 1 cm, the outcome (price) will increase by $0.20 on average.
CONCEPT
Interpreting Intercept and Slope 4
Which of the following scatterplots shows an outlier in both the x- and y-direction?
RATIONALE
To have an outlier in the x-direction and y-direction the outlier must be outside of the range of y data and outside the range of x-data. This outlier is below in the y-direction and to the left in the x-direction.
CONCEPT
Outliers and Influential Points 5
A correlation coefficient between average temperature and ice cream sales is most likely to be .
between 1 and 2
between 0 and –1
between –1 and –2
between 0 and 1
RATIONALE
In general as temperature increases, tastes for ice cream goes up. So the correlation should be positive, which would be between 0 and 1.
CONCEPT
Positive and Negative Correlations 6
Which statement accurately describes the data's form, direction, and strength from the scatterplot below?
Form: Linear Direction: Positive Strength: Weak
Form: Linear Direction: Negative Strength: Weak
Form: Linear Direction: Positive Strength: Moderate
Form: Linear Direction: Negative Strength: Moderate
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, even though the direction is clear, the data points are less clustered in a line or curve, so the strength is moderate.
CONCEPT
Describing Scatterplots 7
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?
5 baskets
9 baskets
3 baskets
8 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 8
For this scatterplot, the r2 value was calculated to be 0.9382.
Which of the following set of statements is true?
About 94% of the variation in beach visitors can be explained by a positive linear relationship with daily temperature. The correlation coefficient, r, is 0.969.
There is no strong correlation in the linear association between beach visitors and daily temperatures. The correlation coefficient, r, is 0.880
About 94% of the variation in daily temperature can be explained by a positive linear relationship with beach visitors. The correlation coefficient, r, is 0.880
About 94% of the variation in beach visitors is explained by a negative linear relationship with daily temperatures. The correlation coefficient, r, is 0.969.
RATIONALE
The coefficient of determination measures the percent of variation in the outcome, y, explained by the regression. So a value of 0.9382 tells us the regression with temperature, x, can explain about 94% of the variation in visitors, y.
We can also note that r = .
CONCEPT
Coefficient of Determination/r^2 9
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?
210
240
350
470
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 10
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 = 0.60
y-intercept = 7.16
Slope = 0.60
y-intercept = 1.22
Slope = 0.54
y-intercept = 1.71
Slope = 0.54
y-intercept = 1.22
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:
y with hat on top equals b subscript 0 plus b subscript 1 x
We know the slope, b subscript 1, and we can use the mean of x and the mean of y for the variables x and y with hat on top to solve for the y-intercept, b subscript 0.
CONCEPT
Finding the Least-Squares Line 11
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.
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.
$0.13 / lb.
$7.75 / lb.
$4.00 / lb.
$6.25 / lb.
RATIONALE
In order to get slope, we can use the formula: s l o p e equals fraction numerator y 2 minus y 1 over denominator x 2 minus x 1 end fraction.
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 12
The scatterplot below charts the performance of an electric motor.
Which answer choice correctly indicates the explanatory variable and the response variable of the scatterplot?
Explanatory variable: Voltage Response variable: Electric motor
Explanatory variable: Rotation Response variable: Electric motor
Explanatory variable: Voltage Response variable: Rotation
Explanatory variable: Rotation Response variable: Voltage
RATIONALE
The explanatory variable is what is along the horizontal axis, which is voltage. The response variable is along the vertical axis, which is speed of rotation.
CONCEPT
Explanatory and Response Variables 13
Shawna finds a study of American men that has an equation to predict weight (in pounds) from height (in inches): ŷ = -210 + 5.6x. Shawna's dad’s height is 72 inches and he weighs 182 pounds.
What is the residual of weight and height for Shawna's dad?
11.2 pounds
809.2 pounds
-11.2 pounds
193.2 pounds
RATIONALE
Recall that to get the residual, we take the actual value - predicted value. So if the actual height of 72 inches and the resulting actual weight is 182 pounds, we simply need the predicted weight. Using the regression line, we can say:
The predicted weight is 193.2 pounds. So the residual is:
CONCEPT
Residuals 14
Which statement about correlation is FALSE?
The correlation of a data set can be positive, negative, or 0.
Correlation is the degree to which the two variables of a data set resemble each other.
Correlation is used to define the variables of only non-linearly related data sets.
Correlation between the variables of the data set can be measured.
RATIONALE
Recall that correlation is used for linear association between 2 quantitative variables, NOT for non-linearly related variables.
CONCEPT
Correlation 15
Which of the following is NOT a guideline for establishing causality?
Take into consideration all the other possible causes.
Look for cases where correlation exists between the variables of a scatterplot.
Keep all variables the same to get duplicate results.
Perform a randomized, controlled experiment.
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.
CONCEPT
Establishing Causality 16
Which of the following statements is TRUE?
If the two variables of a scatterplot are strongly related, this condition implies causation between the two variables.
A correlation of 1 or -1 implies causation.
A high correlation is insufficient to establish causation on its own.
Only a correlation equal to 0 implies 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. Additional information is required to know something is causal, like seeing the association validated in an experimental design.
CONCEPT
Correlation and Causation 17
Which of the following scatterplots shows a correlation affected by an influential point?
RATIONALE
An influential point will influence correlation that doesn't lie in the range of the other data. This graphs shows an outlier that is above the other data and lower in the x-direction.
CONCEPT
Cautions about Correlation 18
Alice reads a scatterplot that shows data for nine schools. It relates the percentage of students receiving free lunches to the percentage of students wearing a bicycle helmet. The plot shows a strong negative correlation.
Alice recalls that correlation does not imply causation. In this example, Alice sees that increasing the percentage of free lunches would not cause children to use their bicycle helmets less.
Identify the confounding variable that is causing Alice's observed association.
The number of free lunches available
School funding
The number of bike helmets available
Parents' income
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 for parents is positively related to owning a bike helmet and this higher income would mean less free school lunches, this variable would help explain why we see this association.
CONCEPT
Correlation and Causation
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