# inference summary ```Inference Procedure Summary – AP Statistics
Procedure
Confidence
Interval for
mean &micro; when
given σ
Hypothesis Test
for mean &micro;
when given σ
(Ho: &micro; = &micro;o)
CI for mean &micro;
when σ is
unknown
Formula
x &plusmn; z*
z=
σ
n
Conditions
One Sample Mean and Proportion
1. SRS
2. Given value of population
standard deviation σ
3. Population distribution is
normal (if not stated, use CLT
as long as n is large)
Calculator Options
x − &micro;o
σ
SAME AS ABOVE CI
n
*Can also find p-value using 2nd-Distr
normalcdf(lower, upper, mean, sd)
x &plusmn;t*
s
n
with df = n - 1
1. SRS
2. Using value of sample
standard deviation s to estimate
σ
3. Population distribution is
given as normal OR n &gt; 40
(meaning t procedures are
robust even if skewness and
outliers exist) OR 15 &lt; n &lt; 40
with normal probability plot
showing little skewness and no
extreme outliers OR n &lt; 15
with npp showing no outliers
and no skewness
Inference Procedure Summary – AP Statistics
Test for mean &micro;
when σ is
unknown
(Ho: &micro; = &micro;o)
x − &micro;o
s
n
t=
SAME AS ABOVE CI
with df = n - 1
*Can also find p-value using 2nd-Distr
tcdf(lower, upper, df)
CI for
proportion p
Test for
proportion p
(Ho: p = po)
pˆ &plusmn; z *
z=
pˆ (1 − pˆ )
n
pˆ − p o
po (1 − po )
n
1. SRS
2. Population is at least 10
times n
3. Counts of success nˆp and
failures n(1 − pˆ ) are both at
least 10 (these counts verify the
use of the normal
approximation)
1. SRS
2. Population is at least 10
times n
3. Counts of success np o and
failures n(1 − p o ) are both at
least 10 (these counts verify the
use of the normal
approximation)
*Can also find p-value using 2nd-Distr
normalcdf(lower, upper, mean, sd)
Inference Procedure Summary – AP Statistics
CI for mean
&micro;1-&micro;2 when σ is
unknown
Two Sample Means and Proportions
1. Populations are independent
2. Both samples are from SRSs
3. Using value of sample
standard deviation s to estimate
σ
s12 s 22
4. Population distributions are
+
( x1 − x 2 ) &plusmn; t *
given as normal OR n1 + n2 &gt;
n1 n2
40 (meaning t procedures are
with conservative
robust even if skewness and
df = n – 1 of
outliers exist) OR 15 &lt; n1 + n2
smaller sample
&lt; 40 with normal probability
plots showing little skewness
and no extreme outliers OR n1
+ n2 &lt; 15 with npps showing no
outliers and no skewness
t=
Test for mean
&micro;1-&micro;2 when σ is
unknown
(Ho: &micro;1 = &micro;2)
( x1 − x 2 )
s12 s 22
+
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
CI for
proportion
p1 – p2
( pˆ 1 − pˆ 2 ) &plusmn; z *
z=
Test for
proportion
p1 – p2
1. Populations are independent
2. Both samples are from SRSs
3. Populations are at least 10
pˆ 1 (1 − pˆ 1 ) pˆ 2 (1 − pˆ 2 ) times n
+
4. Counts of success n1 p̂1 and
n1
n2
n2 p̂ 2 and failures n1 (1 − pˆ 1 ) and
n2 (1 − pˆ 2 ) are all at least 5
(these counts verify the use of
the normal approximation)
( pˆ 1 − pˆ 2 )
1
1
pˆ (1 − pˆ ) +
 n1 n2
X + X2
where pˆ = 1
n1 + n2



1-3 are SAME AS ABOVE CI
4. Counts of success n1 pˆ and
n2 pˆ and failures n1 (1 − pˆ ) and
n2 (1 − pˆ ) are all at least 5
(these counts verify the use of
the normal approximation)
*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
Independence – 1 sample, 2 variables
Homogeneity – 2 samples, 2 variables
1. All expected counts are at
least 1
2. No more than 20% of
expected counts are less than 5
*Can also find p-value using 2nd-Distr
x2cdf(lower, upper, df)
Slope
b &plusmn; t * s b where sb =
CI for β
and s =
s
∑ (x − x)
1
( y − yˆ ) 2
∑
n−2
2
1. For any fixed x, y varies
according to a normal
distribution
2. Standard deviation of y is
same for all x values
with df = n - 2
Test for β
t=
b
with df = n – 2
sb
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 commonly used variables
Variable
&micro;
σ
x
s
z
Meaning
population mean mu
population standard deviation sigma
sample mean x-bar
sample standard deviation
test statistic using normal distribution
Variable
CLT
SRS
npp
p
p̂
z*
critical value representing confidence
level C
test statistic using t distribution
t*
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
critical value representing confidence level C
n
sample size
t
Matched Pairs – same as one sample procedures but one list is created from the difference of two matched lists (i.e. pre and post test
scores of left and right hand measurements)
Conditions – show that they are met (i.e. substitute values in and show sketch of npp) ... don’t just list them
```
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