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Sierra College – Math 13
Spring 2009 – Class 30/
Today: Sections 11-3; 10-1/10-
Assignment: 11-3 {1, 3, 7, 9, 13, 17} 10-2 {1, 3, 5, 7, 9, 13, 17, 19, 23} Next: Sections 10-2/10-
Instructor: John Burke
E-mail: [email protected] Web Page: http://math.sierracollege.edu/Staff/JohnBurke/
Telephone: 916 337-
Office hours: (V-307) MW 2:35-5:00; M 2:45-3:45 (official)
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11-3 Contingency Tables:
Independence and Homogeneity
A contingency table (or two-way frequency table ) is a table in which frequencies correspond to two variables. (One variable is used to categorize rows, and a second variable is used to categorize columns.)
- Example : Titanic passengers categorized as (Survived, Died) by (Men, Women, Boys, and Girls).
- Example : Male survey respondents to an abortion rights question categorized as (Agree, Disagree) by (Male Interviewer and Female Interviewer). - Example : Students categorized as (Male and Female) by (Non-Smoker, Smoker). - Example : Respondents to a question about use of the TV remote categorized as (Male and Female) by (Often, Sometimes, or Almost Never).
In these cases we are looking to determine a dependency relationship between the row variable and column variable, though it is important to note that dependency does NOT establish causality.
11-3 Contingency Tables:
Independence and Homogeneity
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11-3 Contingency Tables:
Independence
A test of independence tests the null hypothesis
H 0 : There is no association between the row variable and the column variable; i.e. the row and column variables are independent. H 1 : The variables in question are not independent.
χ^2 Test for Independence (assumptions)
- The sample data are randomly selected.
- For every cell, the expected frequency is at least 5.
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χ^2 Test for Independence
Test Statistic :
Critical Values :
- The critical values are found in Table A-4 by using degrees of freedom = (r – 1)(c – 1) , where r is the number of rows and c is the number of columns.
- In a test of independence with a contingency table, the critical region is located in the right tail only.
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2 (^ O^ E )
E
Relationships Among Components in Independence Hypothesis Test
Compare the observed (O) values to the corresponding expected (E) values.
Small X^2 value means large p-value Large X^2 value means small p-value
Fail to reject independence
Reject independence
O s and E s are far apart
O s and E s are close
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In a test of homogeneity , we test the claim that different populations have the same proportions of some characteristic.
Example : Male survey respondents to an abortion rights question categorized as (Agree, Disagree) by (Male Interviewer and Female Interviewer).
χ^2 Test for Homogeneity
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χ^2 Test for Homogeneity
Example : Male survey respondents to an abortion rights question categorized as (Agree, Disagree) by (Male Interviewer and Female Interviewer).
- Use a 0.05 significance level.
- H 0 : The proportions of agree/disagree responses are the same for the subjects interviewed by men and the subjects interviewed by women.
- H 1 : The proportions are different.
Men who disagree
Men who agree 240 92
560 308
Man Women
10-1 / 10-3 Correlation and
Regression
In Chapter 10, we examine relationships between paired quantitative data.
We use collected data to
- Observe a pattern (correlation – 10-2)
- Mathematically model the pattern (regression – 10-3)
- When appropriate, use the mathematical model to make predictions.
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Chapter 10 Problem:
Can We Predict the Time of the Next Eruption of Old Faithful?
Is there a relationship between any two variables?
Can we predict how long it will be to the next eruption based upon duration, interval before, or height?
Height (L 4 )* 140 110 125 120 140 120 125 150
Interval After Eruption (L 3 )* 92 65 72 94 83 94 101 87
Interval Before Eruption (L 2 )* 98 90 92 98 93 105 81 108
Duration (L 1 )* 240 120 178 234 235 269 255 220
Eruptions of the Old Faithful Geyser
- Enter the data in your calculator/StatDisk
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10-2 Correlation
Paired sample data is sometimes called bivariate data.
A correlation exists between two variables when one of them is related to the other in some way.
We can often see if a relationship exists by using a scatterplot (or scatter diagram ), a graph in which the paired (x, y) sample data are plotted with each pair represented as a single point.
Assumptions : we will consider only linear relationships, which means that when graphed, the points approximate a straight line. (Recall slope and direction of line.)
Positive Linear Correlation
x x
y y y
x (b) Strong positive
(c) Perfect positive
(a) Positive
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Scatter Plots for the Chapter Problem
StatDisk: Analysis Æ Correlation and Regression
Interval After (L 3 ) vs. Duration (L 1 )
Interval After (L 3 ) vs. Height (L 4 )
Interval After (L 3 ) vs. Interval Before (L 2 )
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Linear Correlation Coefficient
The ( Pearson ) correlation coefficient r measures the strength of the linear relationship between the paired x- and y-quantitative values in a sample.
Assumptions The sample of paired data is a random sample. The pairs of (x, y) data have a bivariate normal distribution.
2 2 2 2
n xy x y r n x x n y y
Notation for r
n = number of pairs of data presented Σ denotes the addition of the items indicated. Σ x denotes the sum of all x values. Σ x^2 indicates that each x score should be squared and then those squares added. ( Σ x )^2 indicates that the x scores should be added and the total then squared. Σ xy indicates that each x score should be first multiplied by its corresponding y score. After obtaining all such products, find their sum. r represents the linear correlation coefficient for a sample ρ (rho) represents the linear correlation coefficient for a population
2 2 2 2
n xy x y r n x x n y y
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Properties of r
The value of r does not change if all values of either variable are converted to a different scale. The value of r is not affected by the choice of x or y. Interchange all x- and y- values and the value of r will not change. r measures the strength of a linear relationship. It is not designed to measure the strength of a relationship that is not linear. r^2 is the proportion of the variation in y that is explained by the linear relationship between x and y.
The value of r is always between -1 and +1 inclusive.
2 2 2 2
n xy x y r n x x n y y
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Table A-
Interpreting r using Table A-6 :
If the absolute value of the computed value of r exceeds the value in Table A-6, conclude that there is a significant linear correlation.
Otherwise, there is not sufficient evidence to support the conclusion of a significant linear correlation.
4 (^56) (^78) 9 (^1011) (^1213) 14 (^1516) (^1718) 19 (^2025) (^3035) 40 (^4550) (^6070) 80 10090
n . .959. .875. . .765. .708. . .641. .606. . .561. .463. . .378. .330. . .269.
. .878. .754. . .632. .576. . .514. .482. . .444. .361. . .294. .254. . .207.
α = .05^ α^ =.
Common Errors Involving Correlation
Causation : It is wrong to conclude that correlation implies causality (Remember eating lobster and its “effect” on pregnancy).
Averages : Averages suppress individual variation and may inflate the correlation coefficient.
Linearity : There may be some relationship between x and y even when there is no significant linear correlation.
