Math 145 - Elementary Statistics Final Exam Summary of ...
Math 145 - Elementary Statistics
Final Exam
Summary of Formulas
? Some Properties of Probability.
1. P (A ¡È B) = P (A) + P (B) ? P (A ¡É B).
2. P (A|B) =
P (A ¡É B)
.
P (B)
? Descriptive Statistics. Let {x1 , x2 , . . . , xn } be a sample of size n, then
P 2 1 P 2
n
n
x ? n ( x)
1X
1 X
SSxx
2
2
1. x? =
xi .
2. s =
(xi ? x?) =
=
.
n i=1
n ? 1 i=1
n?1
n?1
? Discrete Random Variable.
1. ?X = E(X) =
k
X
xi pi = x1 p1 + x2 p1 + x3 p3 + ¡¤ ¡¤ ¡¤ + xk pk .
i=1
2
2. ¦ÒX
= V ar(X) =
k
X
(xi ? ?)2 pi = (x1 ? ?)2 p1 + (x2 ? ?)2 p2 + ¡¤ ¡¤ ¡¤ + (xk ? ?)2 pk .
i=1
n x
? Binomial Distribution. X ¡« bin(n, p) ? P (X = x) =
p (1 ? p)n?x , for x = 0, 1, . . . , n.
x
1. E(X) = np
2. V (X) = np(1 ? p).
? (Continuous) Uniform Distribution. X ¡« unif[c, d]
1. E(X) = 12 (c + d)
? If Z =
2. V (X) =
1
12 (d
?
f (x) =
? c)2 .
1
, for c ¡Ü x ¡Ü d.
d?c
X? ? ?
¡Ì ¡« N (0, 1).
¦Ò/ n
1. The (1 ? ¦Á)100% confidence interval for ? is [X? ? Z ¦Á2 ¡Ì¦Òn , X? + Z ¦Á2 ¡Ì¦Òn ].
Z ¦Á2 ¦Ò 2
2. For a specified margin of error M , the required sample size is n =
.
M
3. To test H0 : ? = ?0 vs. H1 : ? > ?0 , reject the null if Zobs =
X?obs ? ?
¡Ì
> Z¦Á , or if the
¦Ò/ n
p?value= P (Z ¡Ý Zobs ) is ¡Ü ¦Á.
4. To test H0 : ? = ?0 vs. H1 : ? < ?0 , reject the null if Zobs < ?Z¦Á , or if the
p?value= P (Z ¡Ü Zobs ) is ¡Ü ¦Á.
5. To test H0 : ? = ?0 vs. H1 : ? 6= ?0 , reject the null if |Zobs | > Z ¦Á2 , or if the
p?value= 2 ? P (Z ¡Ý |Zobs |) is ¡Ü ¦Á.
? If t =
X? ? ?
¡Ì ¡« t(n?1)
s/ n
1. The (1 ? ¦Á)100% confidence interval for ? is [X? ? t ¦Á2 ¡Ìsn , X? + t ¦Á2 ¡Ìsn ].
X?obs ? ?
¡Ì
> t¦Á .
s/ n
< ?t¦Á .
2. To test H0 : ? = ?0 vs. H1 : ? > ?0 , reject the null if tobs =
3. To test H0 : ? = ?0 vs. H1 : ? < ?0 , reject the null if tobs
4. To test H0 : ? = ?0 vs. H1 : ? 6= ?0 , reject the null if |tobs | > t ¦Á2 .
? If
(n ? 1)S 2
¡« ¦Ö2(n?1) .
¦Ò2
2
1. To test H0 : ¦Ò 2 = ¦Ò02 vs. H1 : ¦Ò 2 > ¦Ò02 , reject the null if Xobs
=
2
(n ? 1)Sobs
> ¦Ö2¦Á .
2
¦Ò0
2
2. To test H0 : ¦Ò 2 = ¦Ò02 vs. H1 : ¦Ò 2 < ¦Ò02 , reject the null if Xobs
=
2
(n ? 1)Sobs
< ¦Ö21?¦Á .
2
¦Ò0
2
2
3. To test H0 : ¦Ò 2 = ¦Ò02 vs. H1 : ¦Ò 2 6= ¦Ò02 , reject the null if Xobs
> ¦Ö2¦Á2 or if Xobs
< ¦Ö21? ¦Á2 .
p? ? p
? If Z = q
¡Ö N (0, 1).
p(1?p)
n
1. The (1 ? ¦Á)100% confidence interval for p is [p? ? Z ¦Á2
q
p?(1?p?)
, p?
n
+ Z ¦Á2
2. For a specified margin of error M , the required sample size is n =
q
p?(1?p?)
].
n
2
Z¦Á/2
[p(1 ? p)]
M2
p? ? p0
3. To test H0 : p = p0 vs. H1 : p > p0 , reject the null if Zobs = q
p0 (1?p0 )
n
¡Ü
2
Z¦Á/2
(0.25)
M2
> Z¦Á .
4. To test H0 : p = p0 vs. H1 : p < p0 , reject the null if Zobs < ?Z¦Á .
5. To test H0 : p = p0 vs. H1 : p 6= p0 , reject the null if |Zobs | > Z ¦Á2 .
? If Z =
(X?1 ? X?2 ) ? (?1 ? ?2 )
q 2
¡« N (0, 1).
¦Ò1
¦Ò22
+
n1
n2
1. The (1 ? ¦Á)100% confidence interval for ?1 ? ?2 is (X?1 ? X?2 ) ¡À Z ¦Á2
q
¦Ò12
n1
¦Ò22
n2 .
+
2. To test H0 : ?1 ? ?2 = d0 vs. H1 : ?1 ? ?2 > d0 , reject the null if Zobs > Z¦Á .
3. To test H0 : ?1 ? ?2 = d0 vs. H1 : ?1 ? ?2 < d0 , reject the null if Zobs < ?Z¦Á .
4. To test H0 : ?1 ? ?2 = d0 vs. H1 : ?1 ? ?2 6= d0 , reject the null if |Zobs | > Z ¦Á2 .
s21
s22
n1 + n 2
2
2
(X?1 ? X?2 ) ? (?1 ? ?2 )
q 2
¡Ö t(k) , where k ¡Ö
? If t =
s1
s22
+
n1
n2
1
n1 ?1
s1
n1
+
2
1
n2 ?1
s22
n2
1. The (1 ? ¦Á)100% confidence interval for ?1 ? ?2 is (X?1 ? X?2 ) ¡À t ¦Á2
2 .
q
s21
n1
+
s22
n2 .
2. To test H0 : ?1 ? ?2 = d0 vs. H1 : ?1 ? ?2 > d0 , reject the null if tobs > t¦Á .
3. To test H0 : ?1 ? ?2 = d0 vs. H1 : ?1 ? ?2 < d0 , reject the null if tobs < ?t¦Á .
4. To test H0 : ?1 ? ?2 = d0 vs. H1 : ?1 ? ?2 6= d0 , reject the null if |tobs | > t ¦Á2 .
(X?1 ? X?2 ) ? (?1 ? ?2 )
q
¡« t(n1 +n2 ?2) , where sp =
? If t =
sp n11 + n12
s
(n1 ? 1)s21 + (n2 ? 1)s22
.
n1 + n 2 ? 2
1. The (1 ? ¦Á)100% C. I. for ?1 ? ?2 is (X?1 ? X?2 ) ¡À t ¦Á2 ? sp
q
1
n1
+
1
n2 .
2. To test H0 : ?1 ? ?2 = d0 vs. H1 : ?1 ? ?2 > d0 , reject the null if tobs > t¦Á .
3. To test H0 : ?1 ? ?2 = d0 vs. H1 : ?1 ? ?2 < d0 , reject the null if tobs < ?t¦Á .
4. To test H0 : ?1 ? ?2 = d0 vs. H1 : ?1 ? ?2 6= d0 , reject the null if |tobs | > t ¦Á2 .
p?1 ? p?2 ? (p1 ? p2 )
? If Z = q
¡Ö N (0, 1).
p1 (1?p1 )
p2 (1?p2 )
+
n1
n2
q
p?1 )
1. The (1 ? ¦Á)100% confidence interval for p1 ? p2 is (p?1 ? p?2 ) ¡À Z ¦Á2 p?1 (1?
+
n1
2. To test H0 : p1 ? p2 = 0 vs. H1 : p1 ? p2 > 0, reject the null if Zobs = q
p?2 (1?p?2 )
n2
(p?1 ? p?2 ) ? 0
p?(1?p?)
n1
Y1 + Y2
, the pooled sample proportion.
n1 + n2
3. To test H0 : p1 ? p2 = 0 vs. H1 : p1 ? p2 < 0, reject the null if Zobs < ?Z¦Á .
where, p? =
4. To test H0 : p1 ? p2 = 0 vs. H1 : p1 ? p2 6= 0, reject the null if |Zobs | > Z ¦Á2 .
+
p?(1?p?)
n2
.
> Z¦Á ,
.
? Chi-Square Statistic. The chi-square statistic is a measure of how much the observed cell counts diverge
from the expected cell counts. The formula for the statistic is
1. Goodness-of-fit Test.
X2
k
X
(observed count ? expected count)2
expected count
i=1
=
k
X
(ni ? n ? pi )2
=
n ? pi
i=1
(1)
¡« ¦Ö2k?1 .
(2)
This X 2 statistic follows approximately the ¦Ö2 distribution with k ? 1 degrees of freedom.
2. Testing equality of several proportions, independence, and homogeneity.
X2
=
=
=
X (observed count ? expected count)2
expected count
all cells
r X
c
X
i=1 j=1
r X
c
X
i=1 j=1
r
(3)
c
X X (nij ? ri ? p?j )2
(Oij ? Eij )2
=
Eij
ri ? p?j
i=1 j=1
(4)
(nij ? ri ? cj /n)2
¡« ¦Ö2(r?1)(c?1) .
ri ? cj /n
(5)
where, ri and cj are the total of the ith row and jth column, respectively. This X 2 statistic follows
approximately the ¦Ö2 distribution with (r ? 1)(c ? 1) degrees of freedom.
? ANOVA Table
Source
DF
Groups
k?1
Error
n?k
Sum of Squares
k
X
ni (x?i ? x?)2
SSG =
SSE =
i=1
k
X
(ni ? 1)s2i
i=1
Total
n?1
SST =
ni
k X
X
Mean Square
M SG =
SSG
k?1
M SE =
SSE
n?k
F
Fobs =
M SG
M SE
(xij ? x?)2
i=1 j=1
Note: SST = SSG + SSE. Under H0 , Fobs ¡« f(k?1,n?k) .
? Equation of the Least-Squares Regression Line . Suppose we have data on an explanatory variable x
and a response variable y for n individuals. The means and standard deviations of the sample data are x? and sx
for x and y? and sy for y, and the correlation between x and y is r. The equation of the least-squares regression
line of y on x is
y? = a? + b?x
with slope
b? =
(¦²xy) ? n1 (¦²x)(¦²y)
SSxy
sy
=r
=
SSxx
sx
(¦²x2 ) ? n1 (¦²x)2
(6)
and intercept
a? = y? ? b?x?
(7)
? The fitted (or predicted) values y?i ¡¯s are obtained by successively substituting the xi ¡¯s into the estimated
regression line: y? = a? + b?xi . The residuals are the vertical deviations, ei = yi ? y?i , from the estimated line.
? The error sum of squares, (equivalently, residual sum of squares) denoted by SSE, is
X
X
X
SSE =
e2i =
(yi ? y?i )2 =
[yi ? (a? + b?xi )]2
X
X
X
=
yi2 ? a?
yi ? b?
xi yi
and the estimate of ¦Ò 2 is
¦Ò? 2 = s2 =
(n ? 1)s2y (1 ? r2 )
SSE
=
.
n?2
n?2
(8)
(9)
(10)
? Linear Correlation. The linear correlation coefficient r measures the strength of the linear relationship
between the paired x? and y?quantitative values in a sample.
P
(xi ? x?)(yi ? y?)
(11)
r = pP
P
(xi ? x?)2 (yi ? y?)2
SSxy
= p
(12)
SSxx SSyy
where,
SSxx
=
SSyy
=
SSxy
=
1
(¦²x)2 = (n ? 1)s2x
n
1
¦²y 2 ? (¦²y)2 = (n ? 1)s2y
n
1
¦²xy ? (¦²x)(¦²y)
n
¦²x2 ?
(13)
(14)
(15)
? The coefficient of determination, denoted by r2 , is the amount of the variation in y that is explained by
the regression line.
r2
=
=
X
SSE
,
where, SST = SSyy =
(yi ? y?)
SST
SST ? SSE
explained variation
=
SST
total variation
1?
(16)
(17)
? Inference for b.
1. Test statistic:
¡Ì
s
b? 1 ? r2
SEb? = ¡Ì
= ¡Ì
sx n ? 1
r n?2
b? ? b
¡« t(n?2)
SEb?
2. Confidence Interval: b? ¡À t¦Á/2 SEb?
? Mean Response of Y at a specified value x? , (?Y |x? ).
1. Point Estimate. For a specific value x?, the estimate of the mean value of Y is given by
??Y |x? = a? + b?x?
2. Confidence Interval. For a specific value x?, the (1 ? ¦Á)100% confidence interval for ?Y |x? is given by
??Y |x? ¡À t¦Á/2;(n?2) SE??
s
where, SE?? = s
1
(x? ? x?)2
+
, and s = sy
n (n ? 1)s2x
r
(n ? 1)(1 ? r2 )
n?2
? Prediction of Y at a specified value x? .
1. Point Estimate. For a specific value x?, the predicted value of Y is given by
y? = a? + b?x?
2. Prediction Interval. For a specific value x?, the (1 ? ¦Á)100% prediction interval is given by
y? ¡À t¦Á/2;(n?2) SEy?
s
1
(x? ? x?)2
where, SEy? = s 1 + +
, and s = sy
n (n ? 1)s2x
r
(n ? 1)(1 ? r2 )
n?2
................
................
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