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A.P. Statistics
Chapter 9 Key points
[9-1]
Definitions
A point estimate of a population characteristic is a single number that is based on sample
data and represents a plausible value of the characteristic
A statistic whose mean value is equal to the value of the population characteristic being
estimated is said to be an unbiased statistic. A statistic that is not unbiased is said to be
biased.
*Given a choice between several unbiased statistics that could be used for estimating a
population characteristic, the best statistic to use is the one with the smallest standard
deviation.
[9-2]
A confidence interval (CI) for a population characteristic is an interval of plausible
values for the characteristic. It is constructed so that, with a chosen degree of confidence,
the value of the characteristic will be captured between the lower and upper endpoints of
the interval.
The confidence level associated with a confidence interval estimate is the success rate of
the method used to construct the interval.
The Large-Sample Confidence Interval for π
The general formula for a confidence interval for a population proportion π when
1. p is the sample proportion from a random sample, and
2. the sample size n is large (np > 10 and n(1 - p) > 10), and
3. if the sample is selected without replacement, the sample size is small relative to
the population size (n is at most 10% of the population size)
is
p  ( z critical value)
p(1  p )
n
The desired confidence level determines which z critical value is used. The three most
commonly used confidence levels, 90%, 95%, and 99%, use z critical values 1.645, 1.96,
and 2.58, respectively.
Note: This interval is not appropriate for small samples. It is possible to construct a
confidence interval in the small-sample case, but this is beyond the scope of this chapter.
The standard error of a statistic is the estimated standard deviation of the statistic.
If the sampling distribution of a statistic is (at least approximately) normal, the bound on
error of estimation, B, associated with a 95% confidence interval is
(1.96) (standard error of the statistic).
Interpretating Confidence Intervals
One hundred 95% confidence intervals for π computed from 100 different random
samples (Asterisks identify intervals that do not include π.)
Interpretation: There is 95% confidence in the method used that the true
mean/proportion of success is between (_____, ______)
*The sample size required to estimate a population proportion π to within an amount B
with 95% confidence is
 1.96 
n   (1   ) 

 B 
2
The value of π may be estimated using prior information. In the absence of any such
information, using π = .5 in this formula gives a conservatively large value for the
required sample size (this value of π gives a larger n than would any other value).
[9-3] Confidence Intervals for population means
CASE I:
The One-Sample z Confidence Interval for μ
The general formula for a confidence interval for a population mean μ when
1. x is the sample mean from a random sample,
2. the sample size n is large (generally n > 30), and
3. σ, the population standard deviation, is known
is
  
x  ( z critical value) 

 n
*If n is small (generally n < 30) but it is reasonable to believe that the distribution of
values in the population is normal, a confidence interval for μ (when σ is known) is:
  
x  ( z critical value) 

 n
CASE II:
The One-Sample t Confidence Interval for μ
The general formula for a confidence interval for a population mean μ based on a sample
of size n when
1. x is the sample mean from a random sample,
2. the population distribution is normal, OR the sample size n is large (generally
n > 30), and
3. σ, the population standard deviation, is unknown
 s 
is x  (t critical value) 

 n
where the t critical value is based on n - 1 df. Appendix Table III gives critical values
appropriate for each of the confidence levels 90%, 95%, and 99%, as well as several
other less frequently used confidence levels.
Important Properties of t Distributions
1. The t curve corresponding to any fixed number of degrees of freedom is bell
shaped and centered at zero (just like the standard normal (z) curve).
2. Each t curve is more spread out than the z curve.
3. As the number of degrees of freedom increases, the spread of the corresponding t
curve decreases.
As the number of degrees of freedom increases, the corresponding sequence of t
curves approaches the z curve.
Comparison of the z curve and t curves for 12 df and 4 df
4.
Let x1, x2 , , xn constitute a random sample from a normal population distribution. Then
the probability distribution of the standardized variable
t
is the t distribution with n - 1 df.
x 
s
n
*The sample size required to estimate a population mean
95% confidence is
 1.96 
n

 B 
to within an amount B with
2
If σ is unknown, it may be estimated based on previous information or, for a population
that is not too skewed, by using (range)/4.
If the desired confidence level is something other than 95%, 1.96 is replaced by the
appropriate z critical value (e.g., 2.58 for 99% confidence).