High School
SAT
ACT
University
MCAT
LSAT
DAT

Correlation and Regression Line

Related Topics

Wize University Statistics Textbook > Simple Linear Regression

Solving for the Regression Line

6 Activities

Wize University Statistics Textbook > Scatterplots, Association, and Correlation

Correlation

2 Activities

If r=−1.00r=-1.00, then 100% of the data points fall exactly on the regression line and the slope cannot be equal or greater than 0.
More Solving for the Regression Line Questions:
Linear Regression
xˉ\bar{x}= 60 yˉ\bar{y}= 4.8
sxs_{x}= 38.08 sys_{y}= 2.59 r=0.71r=0.71
Regression line and Residual plots
Using the data set below, determine the correlation, slope, and intercept of the least squares regression line.
xˉ\bar{x}= 60 yˉ\bar{y}= 4.8
sxs_{x}= 38.08 sys_{y}= 2.59
Practice (Version 1)
Using the data set below, determine the correlation and intercept of the least squares regression line.
∑x=300\sum{x}=300 ∑y=24\sum{y}=24
sx=38.08s_{x}=38.08 sy=2.59s_{y}=2.59
Regression Line
The following data is obtained to assess the relationship between height in cm and shoe size.
The correlation coefficient is 0.93
Which of the following is the equation of the regression line?
Regression Line
One day George wanted to examine the linear relationship between how many hours a student studies (xx) the day before the exam and the exam grade (yy).
He asks 13 random students and collects the following data
r=0.819  sx=3.24  sy=14.24  x‾=8.25  y‾=78.25r=0.819\ \ s_x=3.24\ \ s_y=14.24\ \ \overline{x}=8.25\ \ \overline{y}=78.25
Regression Line
In investigating the relationship between number of people on a person's phone list (xx) and the number of text messages per day (yy), 40 individuals were surveyed and the following information was recorded:
∑i=140xi=700sx=3.5\sum_{i=1}^{40}x_i=700 \quad s_x=3.5
∑i=140yi=3200sy=18.9r=0.67\sum_{i=1}^{40}y_i=3200 \quad s_y=18.9 \quad r=0.67
Regression Line
The equation y^=3+0.025x\hat y=3+0.025x is used to predict the amount of weight lost yy (in kg) by a group of individuals in a study based on the amount of their daily exercise xx (in minutes). Given that 1kg≈2.2lb1kg\approx2.2lb, suppose we want to change the amount of weight lost to pounds (lb) in this equation, which of the following will be the correct linear regression line?
Slope of a Regression Line
Suppose we want to see if there is a relationship between the number of hours a student studies the day before their exam (x) and their exam grade (y). We randomly sample 8 students and record our results, the correlation coefficient is 0.819.
If a student studies an additional 2 hours the day before their final exam, by how much can they expect their mark to increase?
Linear Regression
Here's the data and summary of the weekly earnings and amount of coffee purchased by 5 different students:
xˉ\bar{x}= 60 yˉ\bar{y}= 4.8
sxs_{x}= 38.08 sys_{y}= 2.59 r=0.71r=0.71
Regression Line: Changing Units of $x$
The equation y^=35+1.50x\hat y=35+1.50x is used to predict the hourly wage y of someone with x years of experience.
Suppose that x is changed to months of experience. Which of the following will be the correct regression line?
Regression line
Suppose we want to see if there is a relationship between the size of a condo unit and the selling price of it. We randomly sampled 32 units:
r=0.762sx=278.87sy=60350.59xˉ=1034.84yˉ=445734.38\begin{array}{c}r=0.762\\s_x=278.87\\s_y=60350.59\\\bar{x}=1034.84\\\bar{y}=445734.38\end{array}
With this linear regression plot
Regression Line
Suppose we want to see if there is a relationship between the size of a condo unit and the selling price of it. We randomly sampled 32 units:
r=0.762sx=278.87sy=60350.59xˉ=1034.84yˉ=445734.38\begin{array}{c}r=0.762\\s_x=278.87\\s_y=60350.59\\\bar{x}=1034.84\\\bar{y}=445734.38\end{array}
With this linear regression plot
Regression Line
Suppose we want to see if there is a relationship between the size of a condo unit and the selling price of it. We randomly sampled 32 units:
r=0.762sx=278.87sy=60350.59xˉ=1034.84yˉ=445734.38\begin{array}{c}r=0.762\\s_x=278.87\\s_y=60350.59\\\bar{x}=1034.84\\\bar{y}=445734.38\end{array}
With this linear regression plot
Regression Line: Calculation
One day George wanted to examine the linear relationship between how many hours a student studies (xx) the day before the exam and the exam grade (yy).
He asks 13 random students and collects the following data
r=0.819 sx=3.24 sy=14.24 x‾=8.25  y‾=78.25r=0.819\ \quad s_x=3.24\ \quad s_y=14.24\ \quad\overline{x}=8.25\ \ \overline{y}=78.25
Regression Line: Challenging Use of Slope
Suppose we want to see if there is a relationship between the number of hours a student studies the day before their exam (x) and their exam grade (y). We randomly sample 8 students and record our results. The correlation coefficient is 0.819.
If a student studies an additional 2 hours the day before their final exam, by how much can they expect their mark to increase?
Regression Line: Changing Units of X
The equation y^\hat{y}=35+1.50xx is used to predict the hourly wage y of someone with x years of experience. Suppose that x is changed to months of experience. Which of the following will be the correct regression line?
Regression Line: Changing Units of Y Example
Problem
EThe equation weightLost^\widehat{weight Lost}=3+0.025 (exercise^\widehat{exercise}) is used to predict the amount of weight lost, in kilograms, by a group of individuals in a study based on the average amount of daily exercise, in minutes. Each kilogram is approximately 2.2 pounds. Suppose that we change weight lost to pounds. Which of the following describes the corresponding linear regression line?
Linear Regression
xˉ\bar{x}= 60 yˉ\bar{y}= 4.8
sxs_{x}= 38.08 sys_{y}= 2.59 r=0.71r=0.71
Regression Line
Refer to Question 19.
What is the equation of the regression line?
Regression line and Residual plots
Using the data set below, determine the correlation, slope, and intercept of the least squares regression line.
xˉ\bar{x}= 60 yˉ\bar{y}= 4.8
sxs_{x}= 38.08 sys_{y}= 2.59
Solving for the Regression Line
The equation y^=3+0.025x\hat{y}=3+0.025x is used to predict the amount of weight lost yy, in kilograms, by a group of individuals in a study based on the average amount of daily exercise xx, in minutes. Each kilogram is approximately 2.2 pounds.
Suppose that we change weight lost to pounds. Determine the new linear regression line.
Regression Line
What is the intercept of the regression line?
Regression Line
Suppose Big Joe is added to the scatterplot below. He is 13 years old and has 50% body fat. What will happen to the slope and correlation coefficient?
Correlation and Regression Line
Suppose we want to see if there is a relationship between the number of times a student attended their labs in the entire semester (x) and their final exam grade (y). We randomly sample 8 students and recorded our results.
∑(xi−x‾)(yi−y‾)=264.6\sum_{ }^{ }\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)=264.6
Regression line: Predictions and Residual Plots
True or false?
If the sum of the residuals is equal to 0, then the best-fitting regression line goes through all the data points.
More Correlation Questions:
The correlation between the number of hours of operation and sales in dollars at Pino's Pizza is 0.43. Which of the following is true?
Correlation
The correlation between the number of hours of operation and sales in dollars at Pino's Pizza is 0.43. Which of the following is true?
Correlation
Numbers in a data set are in dollar units ($). Which of the following measures will not be in dollar units?
Correlation
Which of the following statements is/or TURE? (select all that apply)
Correlation
True or False: A correlation coefficient of 0.6 means that 60% of the data points in the scatter plot have a linear relationship.
Predictions and Residual Plots
Suppose the following summary data was given:
x-variable: μx=10, sx=2\mu_x=10,\ s_x=2
y-variable: μy=250, sy=10\mu_y=250,\ s_y=10
Correlation
If we double all the YY data and without changing the XX data, the correlation will not change but the slope will get steeper.
Correlation
If R2=0.86R^2=0.86, then there must be a strong, positive correlation between XX and YY.
Correlation
The correlation between the number of hours of operation and sales in dollars at Pino's Pizza is 0.43. Which of the following is true?
Hypothesis Testing: Correlation
For the following statements, select true or false. Enter T for True, F for False.
Correlation
If we swap xx and yy, what will happen to the correlation coefficient rr?
Correlation
The correlation between midterm grades and final exam grades is r=0.88r=0.88. The correlation between assignment grades and final exam grades is r=0.44r=0.44.
Which of the following is false?
Correlation
In a plot, if we swap xx and yy, rr will not change.
Correlation
r>0r>0; large values of xx are associated with large values of yy.
Correlation
r=−0.91r=-0.91; the correlation is very weak.
Correlation
r<0r<0; increasing xx will cause yy to decrease.
Correlation
r=−1.00r=-1.00; the correlation is perfectly negative.
Correlation
r=2.00r=2.00; the correlation is very strong.
Correlation
r=0.33r=0.33; the correlation is positive, but weak.
Correlation: Coefficient of Determination
True or false?
About 81% of the variation of quantities sold can be explained by price. The correlation between price and sales revenue is r=0.9r=0.9.
Simple Linear Regression: Correlation
True or false? Enter T if True, F if False. (Use capital letters: T, F)
1. If 𝒓 = −𝟏. 𝟎𝟎, then 100% of the data points fall exactly on the regression line and the slope cannot be equal or greater than 0.
T
Correlation
The correlation is 0.56 when the unit for xx is years. What will the correlation coefficient be after xx is changed from years to months of experience? [Enter your answer with 2 decimal places, e.g. 0.36]
Correlation
Correlation rr has no units unless xx and yy have the same units.