# Inference Procedure Summary - AP Statistics

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```					                                   Inference Procedure Summary – AP Statistics
Procedure                    Formula                            Conditions                        Calculator Options
One Sample Mean and Proportion
SRS, normality of population
One sample z-interval for the mean   (plots or CLT) or normality of
Confidence
x , independent observations
Interval for
mean µ when                         σ                 (pop. more than 10x samp.)
x ± z*
given σ                           n                Given value of population
standard deviation σ

One sample z-test for mean
Hypothesis Test
for mean µ
x − µo                   SAME AS ABOVE CI
when given σ                 z=
(Ho: µ = µ o)                     σ
n                                                   *Can also find p-value using 2nd-Distr
normalcdf(lower, upper, mean, sd)

SRS, normality of population
(plots, CLT, or robustness of t-
procedures (see below)) or
normality of x , independent
One sample t-interval for the mean    observations (pop. more than
10x samp.)
CI for mean µ                          s
x ±t*
when σ is                            n                The t procedures are applicable
unknown                                               if a) n > 40, even if skewness
(with df = n – 1)            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                       x − µo
when σ is                     t=
s                       SAME AS ABOVE CI
unknown                                n
(Ho: µ = µ o)
(with df = n – 1)                                                   *Can also find p-value using 2nd-Distr
tcdf(lower, upper, df)

One sample z-interval for a proportion   SRS, normality (successes nˆ  p
and failures n(1 − p ) are both at
ˆ
CI for
proportion p                            p(1 − p)
ˆ     ˆ           least 10), independence of
p± z*
ˆ                              observations (pop. more than
n
10x samp.)

One sample z-test for a proportion     SRS, normality (successes np o
Test for                                                and failures n(1 − p o ) are both
proportion p                          p − po
ˆ                   at least 10), independence of
z=
(Ho: p = po)                         po (1 − po )         observations (pop. more than
10x samp.)
n
*Can also find p-value using 2nd-Distr
normalcdf(lower, upper, mean, sd)
Inference Procedure Summary – AP Statistics
Two Sample Means and Proportions

from distinct populations,
normality (plots, CLT, or see
Two-sample t-interval for difference in     below), obs. within each sample
two means                       are independent (each pop. more
than 10x its respective samp.)

CI for mean
2
s12 s 2         The t procedures are applicable if
( x1 − x 2 ) ± t *      +
µ 1-µ 2 when σ is                               n1 n2           a) n1 + n2 > 40, even if skewness
unknown                                                     and outliers exist; b) 15
with conservative                < n1 + n2 < 40, with normal
df = n – 1 of                  probability plot showing little
smaller sample                  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

( x1 − x 2 )
Test for mean                   t=
µ 1-µ 2 when σ is                              2
s12 s 2
unknown                                 +                       SAME AS ABOVE CI
n1 n2
(Ho: µ 1 = µ 2)
with conservative                                                     *Can also find p-value using 2nd-Distr
df = n – 1 of                                                          tcdf(lower, upper, df) where df is
smaller sample                                                        either conservative estimate or value
using long formula that calculator does
automatically!
Inference Procedure Summary – AP Statistics

Two-sample z-interval for difference in          from distinct populations,
two proportions                                                      ˆ
normality (successes n1 p1 and
CI for                                                       n2 p 2 and failures n1 (1 − p1 ) and
ˆ                         ˆ
proportion                        p1 (1 − p1 ) p 2 (1 − p 2 ) n (1 − p ) are all at least 5),
ˆ       ˆ     ˆ       ˆ
p1 – p2    ( p1 − p 2 ) ± z *
ˆ    ˆ                         +                 2       ˆ2
n1           n2        obs. within each sample are
independent (each pop. more
than 10x its respective samp.)

Two-sample z-test for difference in two
proportions
SAME AS ABOVE CI, except
when assessing normality, we
( p1 − p 2 )
ˆ    ˆ
Test for             z=                                      use the pooled proportion p   ˆ
proportion                           1   1                  (see at left) to check that the
p(1 − p) +
ˆ     ˆ             

p1 – p2                             n1 n2                                        ˆ
counts of success n1 p and
*Can also find p-value using 2nd-Distr
n2 p and failures n1 (1 − p) and
ˆ                      ˆ             normalcdf(lower, upper, mean, sd)
X1 + X 2               n2 (1 − p) are all at least 5
ˆ                                where mean and sd are values from
where p =
ˆ                                                                        numerator and denominator of the
n1 + n2
formula for the test statistic
Inference Procedure Summary – AP Statistics
Categorical Distributions

(O − E ) 2
χ =∑
2
E                            homogeneity)
2. All expected counts are at
Chi Square Test      G. of Fit – 1 sample, 1 variable           least 1
Independence – 1 sample, 2 variables          3. No more than 20% of
Homogeneity – 2 or more samples, 2            expected counts are less than 5
variables

*Can also find p-value using 2nd-Distr
x2cdf(lower, upper, df)
Slope
t-interval for regression slope
1. Independent observations
s            2. Linear relationship between x
b ± t *SEb where SEb =
∑ (x − x )   2   and y
3. For any fixed x, y varies
CI for β
according to a normal
1                             distribution
and s =
n−2
∑ ( y − y) 2
ˆ
4. Standard deviation of y is
same for all x values
with df = n – 2

t-test for regression slope
Test for β                  b                                     SAME AS ABOVE CI
t=        with df = n – 2
SE b
*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   Meaning                                       Variable   Meaning
µ       population mean mu                             CLT       Central Limit Theorem
σ       population standard deviation sigma            SRS       Simple Random Sample
x       sample mean x-bar                               npp      Normal Probability Plot (last option on stat plot)
s       sample standard deviation                        p       population proportion
z       test statistic using normal distribution         ˆ
p       sample proportion p-hat or pooled proportion p-hat for two sample
procedures
z*      critical value representing confidence            t*     critical value representing confidence level C
level C
t      test statistic using t distribution               n      sample size

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). The differences are what need to meet the inference conditions.

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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