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Comparing k Populations
Means – One way Analysis of Variance (ANOVA)
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The F test – for comparing k means
Situation We have k normal populations Let mi and s denote the mean and standard deviation of population i. i = 1, 2, 3, … k. Note: we assume that the standard deviation for each population is the same. s1 = s2 = … = sk = s
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We want to test against
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A convenient method for displaying the calculations for the F-test
The ANOVA Table A convenient method for displaying the calculations for the F-test
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Anova Table Mean Square F-ratio Between k - 1 SSBetween MSBetween
Source d.f. Sum of Squares Mean Square F-ratio Between k - 1 SSBetween MSBetween MSB /MSW Within N - k SSWithin MSWithin Total N - 1 SSTotal
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To Compute F (and the ANOVA table entries):
1) 2) 3) 4) 5)
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Then 1) 2) 3) 4)
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The c2 test for independence
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Situation We have two categorical variables R and C.
The number of categories of R is r. The number of categories of C is c. We observe n subjects from the population and count xij = the number of subjects for which R = I and C = j. R = rows, C = columns
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Example Both Systolic Blood pressure (C) and Serum Cholesterol (R) were meansured for a sample of n = 1237 subjects. The categories for Blood Pressure are: < The categories for Cholesterol are: <
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Table: two-way frequency
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The c2 test for independence
Define = Expected frequency in the (i,j) th cell in the case of independence.
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Justification - for Eij = (RiCj)/n in the case of independence
Let pij = P[R = i, C = j] = P[R = i] P[C = j] = rigj in the case of independence = Expected frequency in the (i,j) th cell in the case of independence.
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H0: R and C are independent
Then to test H0: R and C are independent against HA: R and C are not independent Use test statistic Eij= Expected frequency in the (i,j) th cell in the case of independence. xij= observed frequency in the (i,j) th cell
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Sampling distribution of test statistic when H0 is true
- c2 distribution with degrees of freedom n = (r - 1)(c - 1) Critical and Acceptance Region Reject H0 if : Accept H0 if :
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Standardized residuals
Test statistic degrees of freedom n = (r - 1)(c - 1) = 9 Reject H0 using a = 0.05
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Hypothesis testing and Estimation
Linear Regression Hypothesis testing and Estimation
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Fitting the best straight line to “linear” data
The Least Squares Line Fitting the best straight line to “linear” data
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The equation for the least squares line
Let
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Computing Formulae:
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Then the slope of the least squares line can be shown to be:
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and the intercept of the least squares line can be shown to be:
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The residual sum of Squares
Computing formula
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Estimating s, the standard deviation in the regression model :
Computing formula This estimate of s is said to be based on n – 2 degrees of freedom
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Sampling distributions of the estimators
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The sampling distribution slope of the least squares line :
It can be shown that b has a normal distribution with mean and standard deviation
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The sampling distribution intercept of the least squares line :
It can be shown that a has a normal distribution with mean and standard deviation
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Estimating s, the standard deviation in the regression model :
Computing formula This estimate of s is said to be based on n – 2 degrees of freedom
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(1 – a)100% Confidence Limits for slope b :
ta/2 critical value for the t-distribution with n – 2 degrees of freedom
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(1 – a)100% Confidence Limits for intercept a :
ta/2 critical value for the t-distribution with n – 2 degrees of freedom
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Example In this example we are studying building fires in a city and interested in the relationship between: X = the distance of the closest fire hall and the building that puts out the alarm and Y = cost of the damage (1000$) The data was collected on n = 15 fires.
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The Data
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Scatter Plot
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Computations
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Computations Continued
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Computations Continued
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Computations Continued
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Least Squares Line y=4.92x+10.28
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95% Confidence Limits for slope b :
4.07 to 5.77 t.025 = critical value for the t-distribution with 13 degrees of freedom
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95% Confidence Limits for intercept a :
7.21 to 13.35 t.025 = critical value for the t-distribution with 13 degrees of freedom
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