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AP Stats Chapter 10

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shape, center and spread of a SAMPLING DISTRIBUTION of a sample proportion
SHAPE- approx. normal if np≥10 and n(1-p)≥10
CENTER- µ(p-hat)=p
SPREAD- σ(p-hat)=sq rt(p(1-p)/n) if sample is no more than 10% of population
what do subscripts denote when comparing 2 proportions/means
which group the statistic/parameter is from
shape, center and spread of an SRS of size n1 from population 1 with proportion of successes p1 and an independent SRS of size n2 from population 2 with proportion of successes p2
SHAPE- when n1p1, n1(1-p1), n2p2, and n1(1-p2) are all at least 10, the sampling distribution of p1-p2 is approximately normal
CENTER- the mean of the sampling distribution is p1-p2. that is, the difference in sample proportions is an unbiased estimator of the difference in population proportions
SPREAD- standard deviation of the sampling distribution of p-hat1 - p-hat2 is sqrt((p1(1-p1)/n1) + ((p2(1-p2)/n2) [on happy fun] as long as each sample is no more than 10% of it's population
CONFIDENCE INTERVALS FOR P1-P2:
-what independent condition gives us
-what normal condition gives us
-formula
-when independent condition is met: Standard deviation becomes standard error (bc we don't know parameters P1 or P2) with same formula as previous definition - on happy fun)
-when normal condition is met: find critical value z* for the given confidence level
-formula: (p-hat1 - p-hat2) ± Z*(standard error)
Conditions for a 2-sample proportion z-interval
-RANDOM: the data are produced by a random sample of size n1 from population 1 and n2 from population 2 or by 2 groups of size n1 and n2 in a randomized experiment
-NORMAL: the counts of successes and failures in each sample or group are all at least 10
-INDEPENDENT: both the samples or groups themselves and the individual observations in each sample or group are independent. When sampling without replacement, check that the 2 populations are at least 10x bigger than the corresponding samples
what must you state when doing a 4-part problem?
-hypotheses: null & alternate
-conditions: data must meet all conditions (random, normal, independent)
-state which type of test/interval you are performing (2 sample/1 sample/paired data, z/t, test/interval)
-show adequate work/ equations
-conclusion in context
what do hypotheses look like for 2 sample tests?
Ho: P1-P2=hypothesized value
Ha: P1-P2>/</≠hypothesized value
general formula for a 2 sample test
statistic-paramter/st dev of statistic
(P1-P2)-0/standard deviation of statistic
Pooled (combined) sample proportion equation
P-hatc=successes in both samples /total individuals in both samples= X1+X2/n1+n2
z=(p-hat1 - p-hat2)-0/ sqrt(p-hatc(1-p-hatc)/n1 + p-hatc(1-p-hatc)/n2
when do you use a pooled test?
Only when we are conducting a 2 sample test statistic for PROPORTIONS
How do you know if a test is one tailed or 2 tailed?
2 tailed when Ha has ≠. Anything otherwise: one tailed
when deciding which test/ interval to use, ask yourself...
is this test...
one/two tailed?
proportions/means?
1-sample/2-sample/paired data?
use a z/t score?
shape, center and spread of difference of two means
SHAPE- approximately normal if the population distribution of x-bar1 - x-bar 2 is normal OR n1>30, n2>30
CENTER- µ1-µ2 -> unbiased estimator
SPREAD- σ(xbar1-xbar2)=sqrt(σ1^2/n1 + σ2^2/n2) as long as each sample is no more than 10% of it's population
what can we do when the independent condition is met for a 2-sample t statistic?
we can find the standard deviation (in this case standard error since we don't usually know the parameter)
xbar1-xbar2=sqrt(S1^2/n1 + S2^2/n2)
what can we do if the normal condition is met for 2 sample t statistic?
find the t-score:
t=(xbar1-xbar2)-(µ1-µ2)/sqrt(S1^2/n1 + S2^2/n2)
what is the conservative approach for determining degrees of freedom?
subtract 1 from each n and use the smaller one - calculator will give a different df and then will give a smaller range for CIs
Confidence interval for 2 sample means
statistic ± critical value(standard error)
(xbar1-xbar2) ± t(sqrt(S1^2/n1 + S2^2/n2)) - where t* is the critical value for confidence level C
what are the conditions for 2 sample means procedures?
RANDOM- the data are produced by a random sample of size n1 from population 1 and n2 from population 2 or by 2 groups of size n1 and n2 in a randomized experiment
NORMAL- both population distributions are normal OR both sample groups sizes are large (n1>30, n2>30)
INDEPENDENT- both samples or groups themselves and the individual observations in each sample or group are independent. When sampling without replacement, check that the 2 populations are at least 10 times as large as the corresponding samples
what do the hypotheses look like for significance tests for difference of 2 means?
Ho: µ1-µ2=hypothesized value
Ha: µ1-µ2>/</≠hypothesized value
for the normal condition, what do you do if ...
-sample size<15
-sample size at least 15
-sample size>30
-sample size<15: use 2-sample t procedures if data in both samples appear close to normal ( no outliers/strong skewness)
-sample size at least 15: 2 sample t procedures can be used except in the presence of strong skewness or outliers
-sample size>30:2 sample t procedures can be used even for clearly skewed data when both samples are large
should you use 2 sample t procedures on paired data?
NO
is it better to have equal sample sizes of 2 groups, or differing sample sizes?
equal
should you use pooled procedures for 2 sample t procedures?
no - only on proportions
can results be generalized to the larger population if randomization is not present?
no