Download Table of all inference procedures (Ch. 10-14)

Inference Procedure Summary – AP Statistics
Procedure
Formula
Confidence
Interval for
mean µ when
given σ
One sample z-interval for the mean
Hypothesis Test
for mean µ
when given σ
(Ho: µ = µ o)
x ± z*
Conditions
One Sample Mean and Proportion
σ
n
SRS from population of interest,
normality of x ( CLT if n>30 or
norm pop via plots), “essentially”
ind. observations (samp < 10% of
pop); Know value of population
stand. deviation σ
One sample z-test for mean
z=
x − µo
σ
x ± tdf *
SAME AS ABOVE CI
n
One sample t-interval for the mean
CI for mean µ
when σ is
unknown
Calculator Options
s
n
(with df = n – 1)
*Can also find p-value using 2nd-Distr
normalcdf(lower, upper, mean, sd)
SRS from population of interest,
normality of x (CLT if n>30 or
norm pop via plots, or robustness
of t-procedures (see below)),
“essentially” independent
observations (samp <10% of pop)
The t procedures are applicable if
a) n > 40, even if skewness and
outliers exist; b) 15 < n < 40, with
normal probability plot showing
little skewness and no extreme
outliers; and c) n < 15 with npp
showing no outliers and no
skewness
Inference Procedure Summary – AP Statistics
One sample t-test for the mean
Test for mean µ
when σ is
unknown
(Ho: µ = µ o)
x − µo
s
n
t=
SAME AS ABOVE CI
(with df = n – 1)
One sample z-interval for a proportion
CI for
proportion p
pˆ ± z *
pˆ (1 − pˆ )
n
One sample z-test for a proportion
Test for
proportion p
(Ho: p = po)
z=
pˆ − p o
po (1 − po )
n
*Can also find p-value using 2nd-Distr
tcdf(lower, upper, df)
SRS from population of
interest, normality of p̂
(successes npˆ ≥ 10 and failures
n(1 − pˆ ) ≥ 10 ), “essentially”
ind. observations (samp <10%
of pop)
SRS from population of
interest, normality of p̂
(successes np0 ≥ 10 and
failures n(1 − p0 ) ≥ 10 ),
“essentially” ind. observations
(samp <10% of pop)
*Can also find p-value using 2nd-Distr
normalcdf(lower, upper, mean, sd)
Inference Procedure Summary – AP Statistics
Two Sample Means and Proportions
Both samples are independent SRSs
from distinct populations, normality*
of x1 − x2 , (requires x1 and x2 to
Two-sample t-interval for difference in
two means
CI for mean
µ 1-µ 2 when σ is
unknown
s12 s22
( x1 − x2 ) ± tdf *
+
n1 n2
with conservative df = n – 1 of
smaller sample
be approx. norm., (use CLT or plots
of x1 and x2), or see note below), obs.
within each sample are independent
(each samp <10% of its pop)
*
The t procedures are applicable if a)
n1 + n2 > 40, even if skewness and
outliers exist; b) 15 < n1
+ n2 < 40,
with normal probability plots showing
little skewness and no extreme
outliers; and c) n1 + n2 < 15 with
npp showing no outliers and no
skewness
Two-sample t-test for difference in two
means
Test for mean
µ 1-µ 2 when σ is
unknown
(Ho: µ 1 = µ 2)
t=
usually = 0
( x1 − x2 ) − ( µ1 − µ2 )
s12 s22
+
n1 n2
with conservative df = n – 1 of
smaller sample
SAME AS ABOVE CI
*Can also find p-value using 2nd-Distr
tcdf(lower, upper, df) where df is either
conservative estimate or value using long
formula that calculator does
automatically!
Inference Procedure Summary – AP Statistics
Two-sample z-interval for difference in
two proportions
CI for
proportion
p1 – p2
( pˆ 1 − pˆ 2 ) ± z *
n1 p̂1 and n2 p̂ 2 and failures
ˆp1 (1 − pˆ 1 ) pˆ 2 (1 − pˆ 2 ) n (1 − pˆ ) and n (1 − pˆ ) are all
1
1
2
2
+
n1
n2
at least 5), obs. within each
sample are independent (each
samp <10% of its pop)
Two-sample z-test for difference in two
proportions
Test for
proportion
p1 – p2
z=
Both samples are independent
SRSs from distinct populations,
normality of pˆ1 − pˆ 2 (successes
( pˆ 1 − pˆ 2 )
1
1 
pˆ (1 − pˆ ) + 
 n1 n2 
where pˆ =
X1 + X 2
n1 + n2
SAME AS ABOVE CI, except
when assessing normality, we
use the pooled proportion p̂
(see at left) to check that the
counts of success n1 pˆ and
n2 pˆ and failures n1 (1 − pˆ ) and
n2 (1 − pˆ ) are all at least 5
*Can also find p-value using 2nd-Distr
normalcdf(lower, upper, mean, sd)
where mean and sd are values from
numerator and denominator of the
formula for the test statistic
Inference Procedure Summary – AP Statistics
Categorical Distributions
χ2 = ∑
Chi Square Test
(O − E ) 2
E
G. of Fit – 1 sample, 1 variable; df =
categories – 1
Independence – 1 sample, 2 variables;
df = ( r − 1)( c − 1)
Homogeneity – 2 or more samples, 2
variables; df = ( r − 1)( c − 1)
1. SRS from population of
interest (independent SRSs for
homogeneity)
2. All expected counts are at
least 1
3. No more than 20% of
expected counts are less than 5
*Can also find p-value using 2nd-Distr
x2cdf(lower, upper, df)
Slope
t-interval for regression slope
CI for β
b ± tdf *SEb where df = n – 2:
s
SEb =
∑ ( x − x )2
and s =
1
( y − yˆ ) 2
∑
n−2
1. Independent observations
2. Linear relationship between x
and y
3. For any fixed x, y varies
according to a normal
distribution
4. Standard deviation of y is
same for all x values
t-test for regression slope
Test for β
t=
b
with df = n – 2
SE b
SAME AS ABOVE CI
*You will typically be given computer
output for inference for regression
Inference Procedure Summary – AP Statistics
Variable Legend – here are a few of the most commonly used variables
Variable
µ
σ
x
s
Meaning
population mean “mu”
population standard deviation “sigma”
sample mean “x-bar”
sample standard deviation
Variable
CLT
SRS
npp
p
z
test statistic using normal distribution
p̂
z*
t
critical value representing confidence
level C
test statistic using t distribution
Meaning
Central Limit Theorem
Simple Random Sample
Normal Probability Plot (last option on stat plot)
population proportion
sample proportion p-hat or “pooled” proportion p-hat for two sample
procedures
t*
critical value representing confidence level C
n
sample size
Matched Pairs – same as one sample procedures but one set of data is created from the difference of two matched lists (i.e. pre and
post test scores of left and right hand measurements). The differences are what need to meet the inference conditions, and notation
reflects this, i.e., we use µdiff , xdiff , σ diff , sdiff , etc. .
Conditions – show that they are met (i.e. substitute values in and show sketch of npp) ... don’t just list them