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Material Type: Exam; Professor: Dennis; Class: Statistical Analysis; Subject: Statistics; University: University of Idaho; Term: Unknown 1989;
Typology: Exams
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CIs and Hypothesis tests for 5 #
Result: a^ 8"# s 5 b^5 chi-square 2
100 1a " !b % CI for 5 #:
a 8 " 2 b s a 8 " 2 bs ,
Î# "" Î#
a! b a! b
where the chi-square percentiles correspond to 8 "2 df.
Hypothesis test:
H :! 5 # œ -
H :a 5 #
Û ß Ü à
Test statistic: ;#^ œ a8"^ -^2 b^5 s
Reject H if 2 df or
! #
""
Î# "" Î#
ÚÝ Þá
Û ß Ý á Ü à
Ÿ a b Ÿ
! ! ! a! b
Inferences about. œ " (^)! # ""B
Result: .s œ "s (^)! 0 "s"Bhas a normal distribution
EŠ "s (^)! 0 "s (^) " B‹ œ " (^)! 0 ""B
VŠ "s (^)! 0 "s (^) " B‹ œ 5 #’ 8 0 B"BW “ 1 a^ "b# BB estimate with 5 s^ #
100 1a " !b % CI for " (^)! 0 ""B:
"s 0 "s B „ > 5 s 0 8 W
! " Î#^
BB
!
1 a b
where the Student's t percentile corresponds to 8 "2 df
V Š ] " ]s^ 80" ‹ œ V Š ]s ‹ 0 Va ]80"b
œ 0 0 8 W
1 a b
œ 0 0 8 W
1 a b 1
(estimate with 5 s^ #)
100 1a " !b % prediction interval for ]80":
"s 0 "s B „ > 5 s^0 8 W
! " Î#^
BB
!
a b
Model evaluation
Model evaluation centers around the residuals or errors:
/ œ C " C 3 3 s 3 œ C " 3 "s^! " s""B 3
As in AOV, the residuals should resemble a sample from a normal distribution with a constant variance. In particular, the variability of the residuals should not depend on C or B, nor should there be any systematic patterns in the signs of the residuals. Common diagnostic plots:
ï normal probability plot of residuals
ï scatter plot of residuals (vertical axis) vs. predicted C values ( Cs^3 's)
Outliers are observations that are not well-described by the model, that is, they seem to occur outside of the usual range of variability predicted by the model. Fitted statistical models should be screened for outliers, and the outliers identified should be investigated. Outliers might be recording errors, or might have resulted from identifiable circumstances different from those common to the remaining data; outliers might also simply reflect inadequacy of the model.
Common measures of influence:
ï DFFITS is the standardized change in the predicted value for C 3 when that observation is dropped from the data set
ï DFBETAS are the standardized changes in the parameter estimates "s (^)! and "s" when the observation is dropped
In both cases, the simple rule of thumb is to suspect an overly influential observation when the measure exceeds 2 in absolute value
