STA215H5 Lecture Notes - Lecture 7: Standard Deviation, Scatter Plot, Simple Linear Regression

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1 Jun 2018

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STA215; Chapter 7

What is Linear Regression?

●Linear Regression is a linear approach to modelling the relationship between the

response variable and one or more explanatory variables.

● The case of one explanatory variable is called simple linear regression

● Example;

○ Suppose we find that how long running shoes last is related to how much they

cost BUT we want to know how long a pair of running shoes are expected to

last if we pay $55?

○ Correlation alone won’t answer this question. - Instead, we need a model!!

○ In particular, we need a model to be able to use one variable to predict the

other.

○ The solution: Linear Regression!

● Regression Line

○ A regression line is a straight line that describes how a response variable, y,

changes as an explanatory variable, x ,changes.

○ We can use a regression line to predict the value of y for a given value of x.

● Equation of Straight Line

○ A straight line relating y (response variable) to x (explanatory variable) has an

equation of the form: y = ax + b

○ Where;

■a is the slope, the amount by which y changes when x increases by

one unit

■b is the intercept and corresponds to the value of y when x=0

● Example;

○ Using our motivating example from slide 3, we expect that how long running

shoes last is related to the cost. Data collected from 100 elementary school

students was used to develop a model for this relationship and the following

was obtained:

■ Amount of time running shoes last (hours) = 15.34 + 35.02 x price of

running shoes (dollars)

○ a) What is the slope of this line? Say in words what this value means.

■ The slope is 35.02. This means on average, the amount of time shoes

last increases by 35.02 hours for every dollar increase in price

○ b) What is the intercept? Explain why the value of the intercept is not

statistically meaningful.

■ The intercept is 15.34. This is the amount of time running shoes that

cost $0 last. Even though this interpretation is valid, such prediction

would not make sense because shoes are not free...unfortunately

○ c) Find the predicted amount of time running shoes that cost $55 will last

■ For shoes that cost $55, we predict they will last: 1,941.44 hours 15.34

+35.02(55) = 1,941.44 hours

● Example;

○ Suppose you have a lucky pencil and only use that pencil everyday. It initially

weighed 5 grams when you first bought it. Its’ weight goes down 0.004 grams

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Document Summary

What is linear regression? response variable and one or more explanatory variables. Linear regression is a linear approach to modelling the relationship between the. The case of one explanatory variable is called simple linear regression. In particular, we need a model to be able to use one variable to predict the other. A regression line is a straight line that describes how a response variable, y, We can use a regression line to predict the value of y for a given value of x. changes as an explanatory variable, x ,changes. A straight line relating y (response variable) to x (explanatory variable) has an equation of the form: y = ax + b. A is the slope, the amount by which y changes when x increases by. B is the intercept and corresponds to the value of y when x=0 one unit.

Related Questions

Researchers measured the blood pressures of 22 students in

two situations: when the students were relaxed and when the students were taking an

important examination. The file

systolic.txt

lists the systolic and diastolic pressures for

each student in each situation. Dataset is available at

systolic_url = "https://mcs.utm.utoronto.ca/~nosedal/data/systolic.txt"

Use R

to answer each of the following questions. You have to show

all your work

to get

full credit.

Answers, even if correct, with no justifications will not receive any

marks

Using R

, compute the change in systolic pressure by subtracting systolic pressure when

relaxed from systolic pressure during the exam; call this variable

Using R

, repeat part a) for diastolic pressure. Call the resulting variable

Using R

, make a scatterplot of

versus

, fit a regression line to the data and add

this line to your plot. Use: 2 c) your last name as the main title for your plot. (In my

case it would be: 2 c) Nosedal).

Important note. Use

as your response variable and label horizontal and vertical axes.

d) What is the fitted regression model? Provide slope and y-intercept

, make a residual plot from the regression in part c). Does this plot call into

question the use of a linear model? Why?

Note. Use:

2 e) your last name

as the main title for your plot. (In my case it would

be:

2 e) Nosedal

f) Note the outlier in the residual plot (and on the scatterplot from part c)).

Using

, delete the outlier from the data set. Then repeat parts c) and e). Note. Use:

2 f )

your last name

as the main title for your plots. (In my case it would be:

2 f ) Nosedal

g) What is the fitted regression model (after the outlier has been removed)? Provide slope

Opeele Kyinkafa sells different used 2005 GM cars in excellent condition. The value or the retail price (Price) of a car is dependent on a variety of characteristics such as number of miles the car has been driven (Mileage), manufacturer of the car (Make) such as Buick, Cadillac, Chevrolet, Pontiac, SAAB, and Saturn. The car price is also influenced by the body type (Type) such as sedan, convertible, coupe, hatchback and wagon. It is also affected by number of cylinders (Cylinders) in the engine, the engine size (Liter), cruise control which is an indicator variable representing whether the car has cruise control (yes) or no otherwise and Sound indicator representing whether the car has upgraded speakers (yes) or no otherwise. All cars in this data set were less than one year old when priced and considered to be in excellent condition. Determine which major variables are dummies and are dynamic? Identify 3 potential unobserved heterogenous factors in this dataset. Display the full descriptive statistics of the data in R. Why do you think some variables are starred (*)? What is the median for “price”, standard deviation for “mileage”, skewness for “cylinder” and mean for “sound”? Test if “price” is normally distributed using histogram, kernel density plot and the shapiro wilk test. Create a red scatterplot matrix of price, mileage, cylinder, liter, make. From your graph, is the correlation between price & mileage positive or negative? What’s the pearson correlation coefficient between price & mileage and between “cylinder” & “liter” correct to 2 d.p Create a correlation matrix of the dataset by finding the apt R command for converting all string/dummy variables From the matrix, identify 2 potentially good predictors? Identify 3 pairs of variables that are mostly multicollinear. Which 2 pairs of variables are least correlated? Identify 3 pairs of variables most highly correlated with Cylinder Generate a simple regression plot of price on mileage with a re fitted line. Run the whole multiple regression model (label the equation as "regprice") & use it to answer the other questions. Specify the regression model for the whole data keeping the exact names of the variables. Write the estimated regression equation. In one sentence, which variables are significant and which are not? Justify Is the regression model justifiable? Explain Interpret the coefficient of Mileage, MakeCadillac, TypeSedan, Cylinder, & Cruiseyes. Compare the price of a car with 130,000 mileage, 6 cylinders, 3.8 liters, with cruise and sound, was manufactured by Pontiac with a convertible body type with that of the price of a car with four cylinders, 120,000 mileage, 3.8 liters, without cruise nor sound, was manufactured by Cadillac with a hatchback body type. Which factor(s) appears to be causing a comparative difference (if any)? Using only the VIF, which one of the pairs of variables selected to be multicollinear may be deleted? Justify?