47 terms

AP Stats Chapter 8

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point estimator
a statistic that provides an estimate of a population parameter
point estimate
the value of that statistic from a sample. ideally, a point estimate is our "best guess" at the value of an unknown parameter
what will an ideal point estimator look like?
no bias, and low variability
the interval from x-bar - 2Sx to x-bar + 2Sx will capture µ how much of the time?
95% of the time
what general form will all confidence intervals have?
estimate ± margin of error
confidence interval
has 2 parts:
an interval calculated from the data, which has the form:
estimate ± margin of error
margin of error
tells us how close the estimate tends to be to the unknown parameter in repeated random sampling
confidence level (C)
the overall success rate of the method for calculating the confidence interval. that is, in c% of all possible samples, the method would yield an interval that captures the true parameter value

we usually choose a confidence level of 90% or higher because we want to be quite sure of our conclusions
what is the most common confidence level
95%
what does it mean to say that we are 95% confident?
95% of all possible samples of a given size from this population will result in an interval that captures the unknown parameter
to interpret a c% confidence level is to say...
we are c% confident that the interval from ___ to ___ captures the actual value of the [population parameter in context]
does the confidence level tell us the chance that a particular confidence interval captures the population parameter?
NO
does the confidence level give us a set of plausible values for the parameter?
Yes
General formula for a confidence interval (in formula packet)
statistic ± (critical value)(standard deviation of statistic)
where the statistic we use is the point estimator for the parameter
what does the margin of error consist of?
(critical value)(σ of statistic)
what does the critical value depend on?
the confidence level and the sampling distribution of the statistic
does greater confidence require a smaller or larger critical value?
larger critical value
what does the standard deviation of the statistic depend on?
the sample size n
when does the margin of error get smaller?
-the confidence level decreases
-the sample size n increases
what are the 3 conditions that must be met before calculating confidence intervals?
RANDOM: the data should come from a well-designed random sample or randomized experiment
NORMAL: the sampling distribution of the statistic is approximately normal - means: n≥30, proportions: np≥10 and n(1-p)≥10 OR population distribution is normal
INDEPENDENT: when sampling without replacement, the sample size n should be no more than 10% of the population size N to use our formula for the standard deviation of the statistic
when does standard deviation become standard error (SE)?
when we use p-hat in place of the true population proportion
what is the formula for standard error?
SE=sq rt[(p-hat)(1-p-hat)/n]
how do you find a critical value?
critical value is just a z-score, so if you wanted to find the critical value for an 80% confidence level, (assuming normal condition is met) we will leave out 10% in each tail. look for a z-score with probability 0.10. the closest value is z=1.28, so the critical value z for an 80% confidence interval is z*=1.28
an approximate level C confidence interval for p is...
p-hat ± z* sq rt[(p-hat)(1-p-hat)/n]
to determine a sample size for a specific margin of error (ME) when we aren't given p...
use p-hat=.5 as a guess. this will give the largest possible estimate
Margin of error inequality to find sample size:
z* sq rt[(p-hat)(1-p-hat)/n] ≤ ME
equation for means confidence interval
x-bar ± z*(σ/sq rt n)
equation to determine a sample size n for means
z*(σ/sq rt n)
when we don't know σ of the population for means, how can we estimate?
using simple standard deviation of the sample Sx
for means, when we use the sample standard deviation, what does the new statistic look like?
it will not be normally distributed - it is now called a t distribution
what do t distributions look like?
symmetric, with a single peak at zero
there is much more area in the tails
equation for t score
t= (x-bar-µ)/(Sx÷sq rt n)
what does t tell us?
like any standardized statistic, t tells us how far x-bar is from its mean µ in standard deviation units
are all t distributions the same?
there is a different t distribution for each sample size, specified by its degrees of freedom
degrees of freedom equation
df=n-1 -> used to find t score. the statistic will have approximately a t(n-1) distribution as long as the sampling distribution is close to normal
how do the shapes of density curves of t distributions compare to standard normal curves?
they are similar in shape
how does the spread of density curves of t distributions compare to standard normal curves?
the spread in t distributions are greater than standard normal curves
as the degrees of freedom increase, what happens to the t density curve?
it approaches the standard normal curve ever more closely
t distributions have more/less probability in the tails and more/less in the center than does the standard normal
more, less
to find the critical value t* in table b...
consult the corresponding df on the left column (n-1) and the entry on the bottom for confidence level C
what is the standard error of a sample mean
Sx/sq rt n
where Sx is the sample standard deviation.
what does standard error of a sample mean tell us?
how far x-bar will be from µ on average in repeated SRSs of sample size n
confidence interval for µ
x-bar ± t*(Sx/sq rt n)
what are the conditions that must be met to calculate a confidence interval for µ?
same as for proportions
RANDOM, NORMAL, INDEPENDENT
what happens if one or more of the conditions is not met?
the confidence interval cannot be calculated
when are t procedures robust and not robust?
robust against non normality, not robust against very strong skewness or outliers
will larger or smaller samples improve the accuracy of critical values from the t distributions when the population is not normal?
larger samples