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AP STATISTICS
LIST OF CONFIDENCE INTERVALS/HYPOTHESIS TESTS
Flash Cards:
Front:
Estimate of the
Population mean, μ
(Standard deviation of
population known)
Back:
Parameter: μ
Statistic: x
Conditions:
1) standard deviation of population is
known
2) n > 30 or population distribution is
normal
3) independence and 10% condition
4) SRS
Formula:
x  z*

n
TI/CALC function: ZInterval
Estimate of Population
mean, μ
(Standard deviation of
population unknown)
Parameter: μ
Statistic: x
Conditions:
1) standard deviation of population is
unknown
2) if n<15, population distribution must
be approximately normal
if 15<n<40 population distribution
must not be too skewed and no large
outliers
if n>40 then no large outliers
3) independence and 10% condition
4) SRS
s
Formula: x  t *
, df = n-1
n
TI/CALC function: TInterval
Estimate of single
population proportion, p
Parameter: p
Statistic: p̂
Conditions:
1) SRS
2) Independence and 10% condition
3) npˆ  10 and n(1  pˆ )  10
pˆ (1  pˆ )
n
TI/CALC function: 1-PROPZInt
Formula:
Estimate of difference
in 2 independent
population means, μ1-μ2
pˆ  z *
Parameter: μ1 – μ2
Statistic: x1  x2
Conditions:
1) 2 independent SRS’s
2) if n<15, population distribution must
be approximately normal
if 15<n<40 population distribution
must not be too skewed and no large
outliers
if n>40 then no large outliers
3) independence and 10% condition
Formula: ( x1  x 2 )  t df*
s12 s 22

n1 n2
df=min(n1-1,n2-1) or from calc.
TI/CALC function: 2-SAMPTINT
Estimate of difference
in 2 population
proportions,
p1-p2
Parameter: p1 – p2
Statistic: pˆ 1  pˆ 2
Conditions:
1) SRSs from 2 independent populations
2) Independence and 10% condition
3) n1 pˆ 1  10, n1 (1  pˆ 1 )  10, n2 pˆ 2  10, n2 (1  pˆ 2 )  10
Formula: ( pˆ 1  pˆ 2 )  z *
pˆ 1 (1  pˆ 1 ) pˆ 2 (1  pˆ 2 )

n1
n2
TI/CALC function: 2-PROPZInt
Estimate of mean
Parameter: μdiff
difference between 2
Statistic: x diff
paired population means, Conditions:
μdiff
1) 2 paired SRS’s
2) if n<15, population distribution must
be approximately normal
if 15<n<40 population distribution
must not be too skewed and no large
outliers
if n>40 then no large outliers
3) independence and 10% condition
Formula:
s diff
, df = n-1
n
TI/CALC function: TInterval
Test of a single
population mean, μ
(standard deviation of
x diff  t *
Parameter: μ
Statistic: x
Null Hypothesis: H0: μ = μ0
population known)
Test of a single
population mean, μ
(Standard deviation of
population unknown)
Conditions:
1) standard deviation of population is
known
2) n > 30 or population distribution is
normal
3) independence and 10% condition
4) SRS
x  0
Test Statistic: z 
/ n
TI/CALC function: ZTEST
Parameter: μ
Statistic: x
Null Hypothesis: μ = μ0
Conditions:
1) standard deviation of population is
unknown
2) if n<15, population distribution must
be approximately normal
if 15<n<40 population distribution
must not be too skewed and no large
outliers
if n>40 then no large outliers
3) independence and 10% condition
4) SRS
x  0
Test Statistic: t 
, df = n-1
s
n
TI/CALC function: TTEST
Test of mean
difference of paired
samples, μdiff
Parameter: μdiff
Statistic: x diff
Null Hypothesis: μdiff = 0
Conditions:
1) standard deviation of population is
unknown
2) if n<15, population distribution must
be approximately normal
if 15<n<40 population distribution
must not be too skewed and no large
outliers
if n>40 then no large outliers
3) independence and 10% condition
4) SRS
xdiff  0
Test Statistic: t 
, df = n-1
s diff
Test of the difference
of two independent
population means, μ1-μ2
Test of a single
population proportion,p
n
TI/CALC function: TTEST
Parameter: μ1 – μ2
Statistic: x1  x2
Null Hypothesis: μ1 = μ2 or μ1 - μ2>0
Conditions:
1) standard deviation of population is
unknown
2) if n<15, population distribution must
be approximately normal
if 15<n<40 population distribution
must not be too skewed and no large
outliers
if n>40 then no large outliers
3) independence and 10% condition
4) 2 independent SRSs
x1  x 2  0
Test Statistic: t 
s12 s 22

n1 n2
df = min(n1-1,n2-1) or from Calculator
TI/CALC function: 2-SAMPTTEST
Parameter: p
Statistic: p̂
Null Hypothesis: H0: p = p0
Test of the difference
in 2 independent
population proportions,
p1-p2
Conditions:
1) SRS
2) Independence and 10% condition
3) np0 > 10 and n(1-p0) > 10
pˆ  p 0
Test Statistic: z 
p 0 (1  p 0 )
n
TI/CALC function: 1-PROPZTEST
Parameter: p1-p2
Statistic: pˆ 1  pˆ 2
Combined (or Pooled) proportion: pˆ c 
x1  x2
n1  n2
Null Hypothesis: H0: p1 = p2 or p1 – p2 > 0
Conditions:
1) 2 SRS from independent
populations
2) Independence and 10% condition
3) n1p1> 10 and n1(1-p1) > 10
and n2p2 > 10 and n2(1-p2) > 10
Test Statistic:
z
pˆ 1  pˆ 2  0
pˆ c (1  pˆ c )(
1
1
 )
n1 n2
TI/CALC function: 2-PROPZTEST
Test if a set of
observed values
matches a set of
expected values (one
variable and one sample)
OR – How well does a
CHI-SQUARE GOODNESS-OF-FIT TEST
Hypotheses:
H0: p1 = __, p2 = ___, ……, pn = ___
Ha: At least 1 proportion is different
Or H0: The population distribution is correct
sample distribution
match the hypothesized
population distribution?
Ha: The population distribution is not correct
Conditions:
1) Data are counts
2) SRS (if you wish to generalize results
to entire population)
3) At least 5 counts in each expected cell
(O  E ) 2
test statistic:  2  
E
df = # categories-1
Calc: Put observed values into L1
Put expected values into L2
SUM(L1 – L2)2/L2
Use χ2cdf
OR – use χ2GOF-TEST (TI/84)
OR – GOODFIT PROGRAM (TI/83)
Test to find if category (Test of Homogeneity) - Hypotheses:
proportions are the
H0: p1=p2= p3=p4, ….,pn=pm
same for 2 or more
Ha: Not all proportions are the same
groups (2 or more
Conditions:
groups – one variable
1) SRSs
independent samples)
2) All Data are counts
3) Expected number in each cell is > 5
(O  E ) 2
Test Statistic:  2  
E
df = (row total –1)(column total –1)
Test for association
(Independence) for 2
categorical variables
(two categorical
variables – one sample)
CALC: Put observed values into MATRIX[A],
χ2-TEST, MATRIX[B] will contain the
expected values
(Test of Independence) Hypotheses:
H0: 2 variables are independent
Ha: 2 variables are not independent
Conditions:
1) SRS
2) Data are counts
3) Expected counts are > 5 in each cell
(O  E ) 2
E
df = (row total –1)(column total –1)
Test Statistic:  2  
CALC: Put observed values into MATRIX[A],
χ2-TEST, MATRIX[B] will contain the
expected values
Test for a relationship
(non-zero slope)
between 2 quantitative
variables
Parameter: β (population slope)
Statistic: b
Null hypothesis: H0: β = 0 (There is no linear
relationship between x and y)
Conditions:
1) Scatterplot looks like a line is a good fit
2) (x, y) is an SRS
3) Equal-Variance Condition – The spread
around the line is nearly constant
throughout. Look at the residual plot
for outliers or a fan-shaped pattern
that indicates this condition is not met.
4) Normality – The distribution of y-values
for each x-value follows a normal model.
Check the residuals for normality by
making a normal probability plot for the
residuals.
Test Statistic: t 
b
s
, SEb=
SE b
( x  x ) 2
df = n-2
s = estimated standard deviation of spread
around the regression line
s
Estimate of the slope
of a regression line
1
( y  yˆ ) 2
n2
s
1
 residual 2
n2
CALC: LINREGTTEST
Parameter: β (population slope)
Statistic: b
Conditions:
1) Scatterplot looks like a line is a good fit
2) (x, y) is an SRS
3) Equal-Variance Condition – The spread
around the line is nearly constant
throughout. Look at the residual plot
for outliers or a fan-shaped pattern
that indicates this condition is not met.
4) Normality – The distribution of y-values
for each x-value follows a normal model.
Check the residuals for normality by
making a normal probability plot for the
residuals.
formula: b  t n*2 SEb
Calc: LINREGTINT
SEb=
s
( x  x ) 2