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Unformatted text preview: escriptive Statistics
foot
height Mean
27.8
71.7 Std. Deviation
1.5497
3.0579 N
32
32 Model Summary
Model
1 R
.758 R Square
.574 Adjusted
R Square
.560 Std. Error of
the Estimate
1.0280 ANOVA
Model
1 Regression
Residual
Total Sum of
Squares
42.74
31.70
74.45 df
1
30
31 Mean Square
42.74
1.06 F
40.45 Sig.
.000001 Coefficients Model
1 (Constant)
height Unstandardized
Coefficients
B
Std. Error
.25
4.33
.38
.06 Standardized
Coefficients
Beta
.758 Also note that: SXX = x x 2 = 289.87 200 t
.06
6.36 Sig.
.954
.000001 a. How much would you expect foot length to increase for each 1‐inch increase in height? Include the units. This is asking about the slope: 0.38 centimeters. b. What is the correlation between height and foot length? r = 0.758 (would you be able to interpret the value of r2? c. Give the equation of the least squares regression line for predicting foot length from height. predicted y = yhat = 0.25 + 0.38(x) d. Suppose Max is 70 inches tall and has a foot length of 28.5 centimeters. Based on the least squares regression line, what is the value of the predication error (residual) for Max? Show all work. predicted y = yhat = 0.25 + 0.38(70) = 26.85
observed y – predicted y = 28.5 – 26.85 = 1.65 e. Use a 1% significance level to assess if there is a significant positive linear relationship between height and foot length. State the hypotheses to be tested, the observed value of the test statistic, the corresponding p‐value, and your decision. Hypotheses: H0:_____1 = 0 _____ Ha:_____1 > 0 _______ p‐value: _0.000001/2= 0.0000005 _ Decision: (circle) Test Statistic Value: ____6.36 _______ Fail to reject H0 Reject H0 Conclusion: Thus it appears there is a significant positive linear relationship
between height and foot lengths for the population of college men represented by
the sample. 201 f. Calculate a 95% confidence interval for the average foot length for all college men who are 70 inches tall. (Just clearly plug in all numerical values.) 1
ˆ yt s
n
* (x x) 2 x i x 2 26.85 (2.04)1.028 1 70 71.7 2 32
289.87 26.85 ± 0.425 (26.425, 27.275) g. Consider the residual plot shown at the right. Does this plot support the conclusion that the linear regression model is appropriate? Yes No Explain: The plot shows a random
scatter in a horizontal band
around 0 with no pattern. Note: on exam, students who said ‘NO, because the variation appears to
change with x’ were marked as ok too. 202 Regression
Standard Error of the Sample Slope Linear Regression Model Population Version: Y x E (Y ) 0 1 x Mean: Individual: y i 0 1 x i i where i is N (0, ) Sample Version: ˆ
Mean: y b0 b1 x Individual: yi b0 b1 xi ei s.e.(b1 ) s S XX s x x 2 Confidence Interval for 1 b1 t *s.e.(b1 ) df = n – 2 t‐Test for 1 To test H 0 : 1 0 t b1 S XY S XX df = n – 2 MSREG MSE df = 1, n – 2 Confidence Interval for the Mean Response ˆ
y t * s.e.(fit) df = n – 2 x x y y x x y x x x x 2 or F Parameter Estimators b1 0 s.e.(b1 ) 2 b0 y b1 x Residuals where s.e.(fit ) s 1 (x x) 2 n
S XX Prediction Interval for an Individual Response ˆ
y t *s.e.(pred) df = n – 2 ˆ
e y y = observed y – predicted y where s.e.(pred) s 2 s.e.(fit ) 2 Correlation and its square S XY r r2 S XX S YY Standard Error of the Sample Intercept SSTO SSE SSREG SSTO
SSTO where SSTO S YY SSE Confidence Interval for 0 y y 2 b0 t *s.e.(b0 ) df = n – 2 t‐Test for 0 To test H 0 : 0 0 SSE where n2 ˆ y y e
2 2 t 1 x2 n S XX Estimate of s MSE s.e.(b0 ) s 203 b0 0 s.e.(b0 ) df = n – 2 Additional Notes A place to … jot down questions you may have and ask during office hours, take a few extra notes, write out an extra practice problem or summary completed in lecture, create your own short summary about this chapter. 204...
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This document was uploaded on 02/25/2014 for the course STATS 250 at University of Michigan.
 Summer '10
 Gunderson
 Statistics, Correlation

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