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QMM 250
FORMULA SHEET
EXAM III
DRAFT
n
Rules of summation: 1. For any constant c,
 c  nc
i 1
n
3.
 (x
i 1
n
n
i 1
i 1
x
i 1
Sample x 
N
2 
 (x
i 1
i
 x )2
N
Standard Deviation of x: Population   
Rules of Variance:
i 1
2
x
i 1
i
n
n
Sample s 2 
 (x
i 1
Sample s  s
i
 x)2
n 1
2
x
n
For constant c, E (c )  c .
For constant c and random variable X, E (cX )  cE ( X ) .
For constants a and b and random variable X, E (a  bX )  a  bE ( X ) .
For random variables X and Y, E ( X  Y )  E ( X )  E (Y ) .
Sample binomial proportion:
Rules of Expectation: 1.
2.
3.
4.
i 1
n
i
N
Variance of x: Population
n
 y i )   xi   y i
i
N
Mean of x: Population  x 
n
 cxi  c xi
2.
p
1. For constant c, Var (c)  0 .
2. For constant c and random variable X, Var(cX )  c 2Var( X ) .
3. For constants a and b and random variable X, Var (aX  b)  a 2Var ( X ) .
4. For independent random variables X and Y,
Var( X  Y )  Var( X )  Var(Y )   X2   Y2 .
Binomial Distribution Parameters:
Standard Normal Random Variable:
Standard Error of the Sample Mean:
E ( X )  n
z
Var ( X )  n (1   )
x  X
X
X 
X
n
X
Confidence Interval for  X , given known  X :
x  z / 2
Confidence Interval for  X , given unknown  X :
x  t / 2,n1
n
sX
n

 (1   ) 
Sampling Distribution for Sample Binomial Proportion: p ~ N   ,
 , (sample size
n


requirements of 𝑛𝜋 ≥ 10 𝑎𝑛𝑑 𝑛(1 − 𝜋) ≥ 10).
Confidence Interval for  :
p  z / 2
p(1  p)
n
2
Z 
Sample Size Necessary to Estimate 𝜋 to within 𝐸∗: n    / 2   .25
 E* 
X  0
, distributed N(0,1) if H0 is true.
/ n
X  0
Test Statistic for population mean,  unknown: T 
, distributed t n 1 if H0 is true.
s/ n
p 0
Test Statistic for binomial proportion (  ): Z 
, distributed N(0,1) if H0 is true.
 0 (1   0 )
n
Test Statistic for population mean,  known: Z 
p  Value  P(test statistic is at least as extreme as the observed value of the test statistic | H0 is true)
c
Sum of Squares Total:
nj
SST   ( yij  y ) 2
j 1 i 1
c
Sum of Squares for Factor A: SSA   n j ( y j  y ) 2
j 1
c
Sum of Squared Errors:
nj
SSE   ( yij  y j ) 2
j 1 i 1
Mean Square Factor A:
Mean Square Error:
ANOVA Test Statistic:
SSA
c 1
SSE
MSE 
nc
MSA
, distributed Fc1,nc if H0 is true.
F
MSE
MSA 
Equation of the Population Regression Model:
yˆ i  b0  b1 xi
Equation of the fitted regression line:
n
Sum of Cross-Products:
yi   0  1 xi   i
SS xy   ( xi  x )( yi  y )
i 1
n
SS xx   ( xi  x ) 2
Sum of Squared Deviations in x:
i 1
n
OLS Slope Coefficient:
b1 
 (x
i
i 1
 x )( yi  y )
n
 (x
i 1
i

 x)2
OLS Intercept Coefficient:
b0  y  b1 x
Total Sum of Squares:
SST   ( yi  y ) 2
SS xy
SS xx
n
i 1
Sum of Squared Residuals:
Mean Square Error:
n
n
i 1
i 1
SSE   ( yi  yˆ i ) 2   ( yi  b0  b1 xi ) 2
MSE 
SSE
n2
n
Regression Sum of Squares: SSR   ( yˆ i  y ) 2
i 1
R 2 (coefficient of determination):
R2 
Standard Error of the Regression:
s yx 
SSR
SSE
1
SST
SST
SSE
 MSE
n2
Standard Deviation of Sample Slope Coefficient:
sb1 
s yx
SS xx
Confidence Interval for Population Slope Coefficient, 1 : b1  t / 2,n2 sb1
Test Statistic for Regression Slope Coefficient: T 
b1
, distributed t n2 if H0 is true.
sb1