Download Confidence Interval: Estimate Multiplier Standard Error

Advanced Placement Statistics
Calculator Instructions
Exploratory Data Analysis
Five-number summary
Stat / Calc / 1-Var-Stat
Histogram
1-list: just set the Stat Plot 3rd icon type
Xlist: L1
Freq: 1 (Make sure Alpha key is off)
Zoom Stat
2-lists: just set the Stat Plot 3rd icon type
Xlist: L1
Freq: L2
PRESS GRAPH after the window
has been set up.
Normal Probability Plot
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
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Calculator Window
Xmin = 1 smaller than smallest data
Xmax = 1 bigger than bigger data
Xscl = range / number of bars (integer)
Ymin = -.5
Ymax = the number of times the mode
appears
Yscl = 1
Stat Plot 6th icon type
Data Axis: X
Mark: choose one
Zoom Stat
Normal Distribution
Probability between two values:
normalcdf(lower, upper, mean, std dev)
normalcdf(lower,upper) Table A
invNorm(prop) z-score from Table A
invNorm(prop,mean,std dev) z - score
1
Shading the Normal Distribution
Window Set up
DRAW (2nd / PRGM) / ClrDraw / ENTER
Xmin = 4 std dev to the left of mean
Xmax = 4 std dev to the right of mean
Xscl = range / 8
Ymin = -.5
Ymax = about .8 (you might have to adjust)
Yscl = about .01 (you might have to adjust)
DISTR (2nd / VARS) / DRAW /
ShadeNorm(lower, upper, mean, std dev)
Binomial Distributions
DISTR / A:
binompdf( n, p, outcome )
Example:
Probability that in 5 tries there are 2
successes when p is .3
binompdf( 5, .3, 2 )
binomcdf( n, p, from 0 to outcome )
Example:
Probability that in 5 tries there are at most
2 successes when p is .3
binomcdf( 5, .3, 2 )
Example:
Probability that in 5 tries there are at least 2
successes when p is .3
1 - binomcdf( 5, .3, 1 )
Example:
Probability that in 5 tries there are at least 1
success when p is .3
P(At least one): 1 – P(none)
1 - binomcdf( 5, .3, 0 )
2
t-Distributions
Probability between two values:
tcdf(lower, upper, degrees of freedom)
Shading the t-Distribution
DRAW (2nd / PRGM) / ClrDraw / ENTER
DISTR (2nd / VARS) / DRAW /
Shade_t(lower, upper, df)

Window Set up
Xmin = -3
Xmax = 3
Xscl =1
Ymin = -.1
Ymax =0.4
Yscl = 0.1
2
Distribution
Probability between two values:
 2 cdf(lower,upper,df)
2
Shading the  -Distribution
Shade  (lower,upper,df)
2
Window Set up
Xmin = 0
Xmax = 14
Xscl =1
Ymin = -.1
Ymax =0.3
Yscl = 0.1
Goodness of Fit
L1 : observed
L2 : expected
List / Math / 5: sum
Two-way Tables
Enter observed counts in Matrix [A] in 2nd
x-1
Expected counts will be placed in matrix B
Stat / Tests
 2 -Test
Scatterplots, Correlation and
3
Regression
Correlation coefficient:
CATALOG ( 2nd zero) / find
DiagnosticsOn / ENTER / ENTER
Regression Line:
Stat / Calc / 8.LinReg(a+bx) L1, L2, Y1
To get Y1: VARS / Y-VARS / Function
Zoom Stat
Residuals Plot:
L1: explanatory
L2: response
Y1: regression line
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Y1(L1) → L3 (Predicted)
L2 – L3 → L4 (Obsvd-Pred)= Res
Plot L1 vs. L4
Zoom Stat
Calculator use for Regression Line
and Inferences
s
SEb 
( x  x ) 2
( x  x ) 2 comes from Stat/1vars on the explanatory list:
Sx n 1
Standard error about the line
Confidence interval and
Hypothesis testing for Regression
Slope: b  t * SE b
comes from Stat/1-vars on the
residuals list. Use the term.
Stat/Test/LinRegTInterval and/or
Stat/Test/LinRegTTest
4
seq(expression,x,from,to)
randInt(from, to, how many)
Example
seq(x2,x,1,100): the first 100 squares
randInt(1, 6, 120):rolling a die 120 times
Inference Procedures with Normal
Distributions
Inference Procedures with tDistributions
Standard Deviation is known
Standard Deviation is unknown
One-sample mean Confidence Interval:
 Stat / Tests / 7: ZInterval
 Stats, sample mean and population
std dev given
 Data for L1
 C-Level
 Calculate
One-sample mean Confidence Interval:
 Stat / Tests / 8: TInterval
 Stats, sample mean and std dev
given
 Data for L1
 C-Level
 Calculate
One-sample mean Test of Significance:
 Stat / Tests / 1: Z-Test
 Stats, mean and std dev given
 Data for L1
 Calculate
One-sample mean Test of Significance:
 Stat / Tests / 2: T-Test
 Stats, mean and std dev given
 Data for L1
 Calculate
Two-sample means Confidence Interval:
 Stat / Tests / 9: 2-SampZInt
 Stats, sample mean and population
std dev given
 Data for L1 and L2
 C-Level
 Calculate
Two-sample mean Confidence Interval:
 Stat / Tests / 0: 2-SampTInt
 Stats, sample mean and std dev
given
 Data for L1 and L2
 C-Level
 Calculate
Two-sample means Test of Significance:
 Stat / Tests / 3: 2-SampZTest
 Stats, sample mean and population
std dev given
 Data for L1 and L2
 Calculate
Two-sample means Test of Significance:
 Stat / Tests / 4: 2-SampTTest
 Stats, mean and std dev given
 Data for L1 and L2
 Calculate
Simulations
5
One-sample proportion Confidence
Interval:
 Stat / Tests / A: 1-PropZInt
 Counts and samples size
 C-Level
 Calculate
Two-sample proportion Confidence
Interval:
 Stat / Tests / B: 2-PropZInt
 Counts and samples size
 C-Level
 Calculate
One-sample proportion Test of
Significance:
 Stat / Tests / 5: 1-PropZTest
 Counts and samples size
 Calculate
Two-sample proportion Test of
Significance:
 Stat / Tests / 6: 2-PropZTest
 Counts and samples size
 Calculate
Formulas and Conditions
Random
Variables
Expected value
Rules with
For means
a and b are
constants
 X   xi p i
 a bX  a  b X
 X Y  Y   X
Variances
 2 X  ( xi  x ) 2 p i
General Rules for variances:
For variances
 2 a bX  b 2 2 X
X and Y are independent random
variables
 2 X Y  2 X   2Y
 2 X Y  2 X   2Y  2 x y
 2 X Y  2 X   2Y  2 x y
 2 X Y  2 X   2Y
Binomial Distribution
Setting
1. Either success or
failure
2. Fixed number of
observations
3. n independent
observations
Binomial
Coefficient
n!
n
 
 k  (n  k )! k!
Binomial Probability
n
p( X  k )    p k (1  p) n  k
k
Binomial Distribution
Mean
X n p
 X  n p(1  p)
6
4. same probability
of success
Normal Approximation for Binomial Distributions
When n  p 10 and n  (1  p ) 10 , the binomial distribution X is approximately normal, N (np, np(1  p) )
Sampling Distribution of
a Sample Proportion
Mean
p̂
Standard Deviation
 pˆ 
p̂ is an unbiased estimator of p
 p̂ gets smaller as n increases
 pˆ  p
p(1  p)
n
Conditions:
Only when the population is at least
10 times as large as the sample.
This formula does not apply when
the sample is a large part of the
population.
It can be approximated
with a normal distribution,
p(1  p)
),
n
N ( p,
Conditions:
when n  p  10 and
n  (1  p ) 10
The normal approximation
improves as the sample size n
increases.
For fixed sample size n, the
normal approximation is most
accurate when p is close to 1
2
and least accurate when p is near
0 or 1.
Sampling Distribution of
a Sample Mean
x
x
is an unbiased estimator of
the population mean 
Mean
X  
Standard Deviation
X 

n
Conditions:
This formula can only be used when
the population is at least 10 times as
large as the sample.
If the sample is an SRS
from a population that has
the normal distribution
with mean  and standard
deviation  , then the
sample mean x has the
normal distribution
 with mean  and
N ( ,
)
n
standard deviation  .
n
The Central Limit Theorem
If an SRS of size n is drawn from any population whatsoever with mean  and standard
deviation  and n is large, then the sampling distribution of the sample mean
distribution N (  ,  ) with mean  and standard deviation  .
n
x
is close to the normal
n
7
Normal Density equation
y
1
e
 2
1  x 
 

2  
2nd/VARS/DRAW
ShadeNorm(0,0,mean,stD)
Set window to be around
the mean
2
Estimate  Multiplier  Standard Error
Multiplier  Standard Error = margin of error
Confidence Interval:
Estimate Hypothesiz ed value
Standard Error
Test Statistic:
Parameter of
Interest
  mean
  st dev
Estimate, hypotheses &
Conditions
Multiplier &
Test with DF
Known variance
X
z-interval
H 0   0
Conditions:
 data are from SRS
 sampling distribution of x approx
normal

x  z * SE
z-test
z
x  0
Standard Error (SE)
SE   x 

n
X
Unknown variance
H 0   0
One sample
t-interval
Conditions:
8

n
 data are from SRS(Very Important)
 sampling distribution of x approx
normal for n<15
 t procedures can be used for n  15 if no
outliers or strong skewness are present
 In case of skewness, t procedure can be
used as long as n  40
x  t * SE
SE  ̂ x 
t-test with (n-1) df
t
s
n
x  0
s
n
1  2
x1  x2
H 0 : 1   2
Conditions:
When the sizes of the two samples are equal
and the two populations being compared
have distributions with similar shapes,
probability values from the t table are quite
accurate for a broad range of distributions
when the samples are as small as n1=n2=5.
When the two population distributions have
different shapes, larger samples are needed.
 SRS’s from two distinct populations
 Independent samples
 Both populations are normally distributed
 Means and std dev are unknown
Unknown variances
Two sample
t-interval
2
SE =
2
s1
s
 2
n1 n2
( x1  x 2 )  t *
t
2
2
s1
s
 2
n1 n2
( x1  x 2 )  ( 1   2 )
2
2
s1
s
 2
n1
n2
df = smaller of
n1-1 and n2-1
Both procedures err on the safe side:
higher P-values and lower confidence
than are actually true.
two-sample t procedures are more robust
than the one-sample t method
x
n
H 0 p  p0
pˆ 
Approx z-interval
pˆ  z SE
*
p
Conditions:
 data are from SRS
 pop at least 10 times as large as the
sample
 for a test , npˆ  10 and n(1  pˆ )  10
Approx z-test z 
pˆ  p0
p0 (1  p0 )
n
SE = ˆ p̂ = pˆ (1  pˆ )
n
for confidence intervals
p0 (1  p0 )
n
for hypothesis testing
SE =  p̂ =
for confidence interval , np 0  10 and
n(1  p0 )  10
9
Sample size given a margin of error
2
 z* 
n    p* 1  p*
m


Where p* is .5
( p1  p2 )
pˆ 
pˆ1  pˆ 2
Confidence Intervals
H 0 : p1  p2
( pˆ 1  pˆ 2 )  z * SE
successes in both samples
totlal successes
pˆ 

n2 pˆ 2  5, n2 (1  pˆ 2 )  5
pˆ 1 (1  pˆ 1 ) pˆ 2 (1  pˆ 2 )

n1
n2
Hypothesis Testing
X1  X 2
n1  n2
Conditions:
 The populations are at least 10
times as large as the samples

n1 pˆ 1  5, n1 (1  pˆ 1 )  5,
SE 
z
pˆ 1  pˆ 2
pˆ (1  pˆ )(
1
1
 )
n1 n 2
Conditions:
 The populations are at least 10
times as large as the samples
n1 pˆ  5, n1 (1  pˆ )  5,


n2 pˆ  5, n2 (1  pˆ )  5
( pˆ 1  pˆ 2 )
x  x 
  1    2 
 n1   n2 
 2 number is define as
Goodness of Fit Test
Multiple
proportions
H0: the actual population proportions are
equal to the hypothesized
Ha: the actual population proportions are
different from the hypothesized
n: number of outcome categories
df: n-1
(O  E ) 2
 
E
2
Where the E counts are calculated with
the proportions from the H0 hypothesis.
Conditions:
 All individual expected counts are
at least 1 and no more than 20% of
the expected counts are less than 5.
 SRS
Test for
10
Homogeneity of populations
H0: p1 = p2 = p3 = …
Ha: not all proportions are
equal
Conditions:
 All individual expected counts are
at least 1 and no more than 20% of
the expected counts are less than 5.
 Multiple SRS’s
Two Way
Tables
 2 number is define as
2  
(O  E ) 2
E
Where the E  rowTotal  columnTotal
TableTotal
Test of
Association/Independence
 2 number is define as
H0: there is no relationship between
two categorical variables
Ha: there is relationship between
two categorical variables
(O  E ) 2
 
E
2
Conditions:
 All individual expected counts are
at least 1 and no more than 20% of
the expected counts are less than 5.
 A single SRS
Where the E  rowTotal  columnTotal
TableTotal
Linear Regression
Correlation Formula


 xi  x  yi 
1

r

n  1  s x  s y
Least Square Regression Line
y 


yˆ  a  bx
with slope
br
Sy
Sx
b
yˆ  a  bx
and intercept
a  y  bx
residuals
11
residuals  y  yˆ
Inference for Regression
Model


Conditions:
* Observations are independent
* True linear relationship
* The standard deviation of the
response about the true lines is
the same everywhere
* The response varies normally about
the true regression line.
Confidence Interval for the slope of the
line:
b  t * SEb
df=n-2
SEb 
s
( x  x ) 2
( x  x ) 2
comes from
Stat/1-vars on
the explanatory
list:
Sx n 1
Standard error about the line
s
Confidence Interval
y
yˆ  t SE̂
df=n-2
*
( y  yˆ ) 2
n2
S(y - yˆ ) 2 comes
from
Stat/1-vars on the
residuals
list.
2
Use the term Sx .
1
( x*  x ) 2
SE ˆ  s

n  (x  x)2
12
ŷ
1
( x*  x ) 2
SE yˆ  s 1  
n  (x  x)2
Prediction Interval
yˆ  t * SE yˆ
Transforming Relationships
Algebra
Defining and using Logarithms
logx = y if and only if by=x
log(response)= a + k log (explanatory)
Properties of Logarithms
log(AB) = log A + log B
log(A/B) = log A – log B
log Xp = p log X
10log(response) = 10a+log(explanatory^k)
Response = 10a
x explanatoryk
13