Chapter 4 Correlation and Regression Understanding Basic Statistics Fifth Edition By Brase and Brase Prepared by Jon Booze

5 | 2 Copyright © Cengage Learning. All rights reserved. Scatter Diagrams A graph in which pairs of points, (x, y), are plotted with x on the horizontal axis and y on the vertical axis. The explanatory variable is x. The response variable is y. One goal of plotting paired data is to determine if there is a linear relationship between x and y.

5 | 3 Copyright © Cengage Learning. All rights reserved. Paired Data (x, y) Important Questions How strong is the linear correlation between x and y? What line best represents the data?

5 | 4 Copyright © Cengage Learning. All rights reserved. How Strong Is the Linear Correlation? Not all relationships are linearly-correlated. Statisticians need a quantitative measure of the strength of the linear association.

5 | 5 Copyright © Cengage Learning. All rights reserved. The Sample Correlation Coefficient r Statisticians use the sample correlation coefficient r to measure the strength of the linear correlation between paired data. 1)r has no units. 2)–1 ≤ r ≤ 1 3)r > 0 indicates a positive relationship between x and y, r < 0 indicates a negative relationship. 4)r = 0 indicates no linear relationship. 5)Switching the explanatory variable and response variable does not change r. 6)Changing the units of the variables does not change r.

5 | 6 Copyright © Cengage Learning. All rights reserved. Interpreting the Value of r r = 0 There is no linear relation for the points of the scatter diagram.

5 | 7 Copyright © Cengage Learning. All rights reserved. Interpreting the Value of r r = 1 or r = –1 There is a perfect linear relation between x and y; all points lie on a straight line.

5 | 8 Copyright © Cengage Learning. All rights reserved. Interpreting the Value of r 0 < r < 1 The x and y values has a positive correlation. As x increases, y tends to increase.

5 | 9 Copyright © Cengage Learning. All rights reserved. Interpreting the Value of r –1 < r < 0 The x and y values have a negative correlation. As x increases, y tends to decrease.

5 | 10 Copyright © Cengage Learning. All rights reserved. Critical Thinking |r| ≈ 1 only implies a linear relationship between x and y. It does not imply a cause and effect relationship between x and y. The values of x and y may both depend linearly on some third lurking variable.

5 | 11 Copyright © Cengage Learning. All rights reserved. Linear Regression Linear Regression - a mathematical technique for creating a linear model for paired data. Based on the “least-squares” criterion of best fit.

5 | 12 Copyright © Cengage Learning. All rights reserved. Least-Squares Criterion

5 | 13 Copyright © Cengage Learning. All rights reserved. Properties of the Regression Equation The point is always on the least-squares line. The slope tells us the amount that y changes when x increases by one unit.

5 | 14 Copyright © Cengage Learning. All rights reserved. Illustration Least-squares linear relationship between caribou and wolf populations:

5 | 15 Copyright © Cengage Learning. All rights reserved. Critical Thinking: Making Predictions We can simply plug in x values into the regression equation to calculate y values. Extrapolation may produce unrealistic forecasts.

5 | 16 Copyright © Cengage Learning. All rights reserved. Coefficient of Determination Another way to gauge the fit of the regression equation is to calculate the coefficient of determination, r 2. 1). Compute r. Simply square this value to get r 2. 2). r 2 is the fractional amount of total variation in y that can be explained using the linear model. 3). 1 – r 2 is the fractional amount of total variation in y that is due to random chance (or possibly due to lurking variables).

Chapter 5 Elementary Probability Theory Understanding Basic Statistics Fifth Edition By Brase and Brase Prepared by Jon Booze

5 | 18 Copyright © Cengage Learning. All rights reserved. Probability Probability is a numerical measure that indicates the likelihood of an event. All probabilities are between 0 and 1, inclusive. A probability of 0 means the event is impossible. A probability of 1 means the event is certain to occur. Events with probabilities near 1 are likely to occur.

5 | 19 Copyright © Cengage Learning. All rights reserved. Probability Events can be named with capital letters: A, B, C… P(A) means the probability of A occurring. –P(A) is read “P of A” –0 ≤ P(A) ≤ 1

5 | 20 Copyright © Cengage Learning. All rights reserved. Probability Assignment Assignment by intuition – based on intuition, experience, or judgment. Assignment by relative frequency – P(A) = Relative Frequency = Assignment for equally likely outcomes

5 | 21 Copyright © Cengage Learning. All rights reserved. Law of Large Numbers In the long run, as the sample size increases, the relative frequency will get closer and closer to the theoretical probability. Example: Toss a coin repeatedly. The relative frequency gets closer and closer to P(tails) = 0.50 Relative Frequency f = number of tails n = number of flips

5 | 22 Copyright © Cengage Learning. All rights reserved. Probability Definitions Statistical Experiment: Any random activity that results in a definite outcome. Event: A collection of one or more outcomes in a statistical experiment. Simple Event: An event that consists of exactly one outcome in a statistical experiment. Sample Space: The set of all simple events.

5 | 23 Copyright © Cengage Learning. All rights reserved. The Sum Rule The sum of the probabilities of all the simple events in the sample space must equal 1.

5 | 24 Copyright © Cengage Learning. All rights reserved. Probability versus Statistics Probability is the field of study that makes statements about what will occur when a sample is drawn from a known population. Statistics is the field of study that describes how samples are to be obtained and how inferences are to be made about unknown populations.

5 | 25 Copyright © Cengage Learning. All rights reserved. Independent Events Two events are independent if the occurrence or nonoccurrence of one event does not change the probability of the other event.

5 | 26 Copyright © Cengage Learning. All rights reserved. Multiplication Rule for Independent Events General Multiplication Rule – For all events (independent or not): Conditional Probability (when ):

5 | 27 Copyright © Cengage Learning. All rights reserved. Meaning of “A and B”

5 | 28 Copyright © Cengage Learning. All rights reserved. Meaning of “A or B”

5 | 29 Copyright © Cengage Learning. All rights reserved. Mutually Exclusive Events Two events are mutually exclusive if they cannot occur at the same time. Mutually Exclusive = Disjoint If A and B are mutually exclusive, then P(A and B) = 0

5 | 30 Copyright © Cengage Learning. All rights reserved. Addition Rules If A and B are mutually exclusive, then P(A or B) = P(A) + P(B). If A and B are not mutually exclusive, then P(A or B) = P(A) + P(B) – P(A and B).

5 | 31 Copyright © Cengage Learning. All rights reserved. Multiplication Rule for Counting This rule extends to outcomes involving three, four, or more series of events.

5 | 32 Copyright © Cengage Learning. All rights reserved. Multiplication Rule for Counting A coin is tossed and a six-sided die is rolled. How many outcomes are possible? a). 8b). 10c). 12d). 18

5 | 33 Copyright © Cengage Learning. All rights reserved. Tree Diagrams Displays the outcomes of an experiment consisting of a sequence of activities. –The total number of branches equals the total number of outcomes. –Each unique outcome is represented by following a branch from start to finish.

5 | 34 Copyright © Cengage Learning. All rights reserved. Four possible outcomes: Red, Red Red, Blue Blue, Red Blue, Blue Probabilities are found by using the multiplication rule for dependent events. Place five balls in an urn: three red and two blue. Select a ball, note the color, and, without replacing the first ball, select a second ball. Urn Example

5 | 35 Copyright © Cengage Learning. All rights reserved. The Factorial n! = (n)(n – 1)(n – 2)…(2)(1), n a counting number By definition, 1! = 1 0! = 1 Example: 5! = 5·4·3·2·1 = 120

5 | 36 Copyright © Cengage Learning. All rights reserved. Permutations Permutation: ordered grouping of objects. Example Permutation: Seats 1 through 5 are occupied by Alice, Bruce, Carol, Dean, and Estefan, respectively.

5 | 37 Copyright © Cengage Learning. All rights reserved. Combinations A combination is a grouping that pays no attention to order. Example Combination: Out of a set of 20 people, Alice, Bruce, Carol, Dean, and Estefan are chosen to be seated.