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**Unformatted text preview: **Regression Distribution of Random Error r I ,1 L 2)
How is e distributed. for each single value ofx? y s: + E)
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r15360+ﬂfxz7+£zb Regression Assumptions Co’wﬁdrnd the mm»: m‘ ranolm cm 4mm 5
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1. At any given value of x, the mean of the distribution of e = 0. 205,730 5”” £56” 2 L4 ,‘g‘a mean ‘1
é Cm: é (Ba Wm +6): 2‘ (50+5’5X1ﬁ5“)
ff”): 5096' 1L3; Y‘IJ'I‘O ((YJ: 80 +8,x 2. For all values of x, the variance of 6 (oz) is constant. an ,4 (pcqf‘u'oﬂ -_
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(7' 1C)”: 0'3 Ce?) 4. e ’3 associated with any 2 observations are independent.
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Ciark — BIT 2406 Page 14 Regression Variance of 6 (0‘2)
The more variability in e, the greater the error of estimation for 9 .
5mm N M Vedanta 0-2 ha; éé-hko, 6741‘ m w'fr'on {or—r iar8€ varfﬁnce or? p {.5 no}— qfcureqf-e
fie/‘46, and gm Population variance of 6(02) is estimated by $2: 2 SSE SSE S 2
degrees of freedom for error n ~ 2 where SSE = XUi * 9i )2
'7'» 522' -.- 76*75’1'0 .- 33,3951" 5'n-2/----'“*‘ (1‘ 5,; Standard Deviation of e(o-) is estimated by s: 5=Jg=% 5: V383675 1 G/‘i‘lo Clark - BIT 2406 Page 15 Regression Hypothesis Testing
Testing the Slope B1 Hypothesis Test on [31
Ho: [51:0 :9 (5,), “51' {4567411 to [Moder/2 H3: [51 at 0
Choose a. ievel C (’0qu CELT,» 14’ Type I 3 rm» ) Test with statistic: .. [’0] .. kw: 51
S. S
B] /\i SSxx Two —+crr‘l~eJ 4‘5""
Decision Rule: I will reject Ho if |tcalci > t% n _ 1
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Clark—BIT 2406 rr'caft, 4:; a} 4/7, r “.2, 0’74 "" +61: id f'fqi'ﬁ 71-0“ n»; ‘4’qu Page 16 /V\~2 Regression Grade vs. Study Time Example Exam # Stud Time Grade on Exam x2
1 4 hr 41 164 16 1661
2 9 hr 63 567 61 3969
3 16 hr 87 1392 256 7569
4 23 hr 96 2254 529 9605
52 269 4377 662 22823 Recall SSE = 76.7306 and S = 6.1940 Two-tailedtestatoc=o.05::7 Ngecl’" “91 I" “add ’ 7mg" *’~ Ho: 61:9 +646 3 3/."0 ~: 7300?? :: 9; {’7'15’
C 7%? Ha: 61“) 55/337 l /Jz—06 +~+4ue ; 12.6% 2 “*1 2 “1°28; 7 “JOB .1. 69/5 7 fix/Z, h-2
(an c640”? ﬂeJ'éC‘F Mr H0 1’ x13 " '. W/fé’d‘dy One-tailedtest(a=0.05)::;" ﬂei'edh Ho‘ [f 4-0,. h: >fd
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X “5 “NF”! l‘o ymdrc‘f“ Y Clark — BlT 2406 Page 17 Grade 1 a’q'IZB? N} Regression Conﬁdence Interval for [31 Want to say: We are 911% conﬁdent that B, lies within the interval 100 (1-r1)% Conﬁdence Interval: ﬂit“ sB -; 3,009”
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’P 4 (“'1 “If-thin fag twang”: Clark — BIT 2406 Page 18 Regression Coefﬁcient of Determination Represents the proportion of vaniation explained by the model. _ .. ~ {Tcn / MOO/b mode]
r2:m :1_SSE 5 2", IV‘df‘tq 6"!“ 7
SSW SSW mm Var/(‘qf-‘M ' 2
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52: f’ ———-—-""76‘7 a [234); Z 0. 7695 3 76(55 /' 3. 6’5“; 7, of +"‘e Varl‘ﬁff‘d"! 6"" G? QXFKQD'IEGI 9y 0 th er Fad-0r; Clark — BIT 2406 Page 19 Regression Coefficient of Correlation The coefﬁcient of oon'elation measures the strength of the linear relationship between x and y. n ’ ‘72 n" I“! {679% 5 my: 3st
r z — (—1 3 rs 1)
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s = o. ‘i' re!
How Regression is Misused
Extrapolation (“.0qu beyond fine (4,54 offer fampxp 0(ch Over-Reliance on Outliers and Extremes fa. chili-err Assume Cause-Effect Relationship diffmhg Ca “re ’2‘ Np" "‘e'ms r"; I“ h.“ H"? Nonsense Models ﬂo j/gpeJ n4" ﬁehﬁoash¢ Clark — BIT 2406 Page 20 Regression Procedure to Develop a Model Step 1 Hypothesize deterministic nent of model
y: 60 +ﬂ! X 't E, 3th: da+ffmr.n;5;r‘5 PEN" + I3. ‘1 Line, 8mp2 Estimate parameters ,, ”
am,“qu 59,3, one, 60 “J :9, 8mp3
Evaluate random error distribution
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(“66:0 W5 € 8mp4 Determine model usefulness H {cream févﬁ' ~0'nt 1‘6"] (+/../ _ .
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Use model To We (Jaci- asf-‘Nﬁ‘ﬁ o‘y Clark — BIT 2406 Page 21 Regression
Regression Testing Review BIT 2406 A set of sample data has the ﬁtted least squares line )7 = 49.1016 - 0.9590)(.
n = 18, SSE = 218.1624, 2x: 346, 2x2 = 7614, By: 552. £y2 =18,032, ny = 9687 I want to test the slope of the least squares line to determine if there is a statistically signiﬁcant
negative relationship between x and y. 1. What is the null hypothesis? The alternative hypothesis?
Halﬁfco Hq” 640 2. What is the decision rule in: = 0.025?
’(5 “F Ho f-F .
t ML <- tan ,1 ;> +wrc < u tater/5‘, '0 41> fag/c <~— 2. 7.20
3. What is the value of the test statistic calculated from the sample data?
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5; gig, :iéqzé, 5synggz—g’ﬂd': ‘ié'iH 4. Can we conclude that x is useful in predicting y with a negative relationship between the two
variables at a = 0.025? kg 5. Whatisthe coefﬁcient of determination?
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V1 6. What is the coefﬁcient of correlation? ~ 7": 7§KV wt: 2?,{9667
low-55w I catamarerer (HQ "ﬂ 7 " .« Lula (“def-fur): "(/03ij :5 ‘— 0'5“??? 7. We are 80% conﬁdent that the true value of the slope lies within what interval?
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r.- f I... Clark — BIT 2406 ...

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