Review 3
Chapter 8
1. A statistic  any quantity computed from values
in a sample (for example, x , s, the sample median,
the sample interquartile range and so on).The
distribution of a statistic is called its sampling
distribution.
A population parameter  any quantity computed
from values in a population (for example, , , the
population median, the population interquartile range
and so on).
 The difference between a statistic and a population
parameter.
(1) A statistic is a sample characteristic, whereas a
population parameter is a population characteristic.
(2) The observed value of a statistic varies from
sample to sample. However, a population parameter
is a fixed number, which is generally unknown.
2. Properties of the sampling distribution of x
Let x denote the mean of the observations in a
random sample of size n from a population having
mean  and standard deviation . Denote the mean
value of the x distribution by  x and the standard
deviation of x distribution by  x . Then the following
rules hold.
Rule 1:  x =
Rule 2:  x =  .
n
Rule 3: When the population distribution is normal,
the sampling distribution of x is also normal
for any sample size n. Thus, the
standardized variable
z
x x
x

x
 / n
has the standard normal (z) distribution.
Rule 4: (Central Limit Theorem) When n is
sufficiently large (n ≥ 30), the sampling
distribution of x is well approximated by a
normal curve, even when the population
distribution is not itself normal. So, the
standardized variable
z
x x
x

x
 / n
has approximately the standard normal (z)
distribution.
3. Properties of the sampling distribution of p
Let p be the proportion of S’s in a random sample of
size n from a population whose proportion of S’s is .
Denote the mean value of p by p and the standard
deviation of p by p. Then the following rules hold.
Rule 1: p = 
Rule 2:  p 
 (1   )
n
Rule 3: (Central Limit Theorem) When n is large and
 is not too near 0 or 1 (n  10 and n(1- )
 10), the sampling distribution of p is
approximately
normal.
Thus,
the
standardized variable
z
pp
p

p 
 (1   ) / n
has approximately the standard normal (z)
distribution.
Chapter 9
4. A point estimate of a population characteristic is a
single number computed from sample data and
represents a plausible value of the characteristic. A
point estimate is obtained by (i) selecting an
appropriate statistic; (ii) computing the value of the
statistic for the given sample.
A statistic whose mean 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 biased.
5. Criteria
statistics
for
choosing
among
competing
a) First we choose an unbiased statistic if there is one.
b) If several unbiased statistics could be used for
estimating a population characteristic, we choose
the one with the smallest standard deviation.
6. Statistics used to estimate some important
population characteristics
Population characteristic Statistic to use
to be estimated
p
Population proportion, 
x
Population mean, 
s2
Population variance, 2
Population standard
s
deviation, 
Population median
Sample median
Unbiasedness
Unbiased
Unbiased
Unbiased
Biased
Unbiased if symmetric
Biased if skewed
7. A confidence interval 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 inside the interval.
The confidence level associated with a confidence
interval estimate is the success rate of the method used
to construct the interval.
The standard error of a statistic is the estimated
standard deviation of the statistic.
If the sampling distribution of a statistic is normal
(approximately), the bound on error of estimation, B,
associated with a confidence interval is
(z critical value)(standard deviation of the statistic).
8. The large-sample confidence interval for 
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)
the general formula for a confidence interval for a
population proportion  is
p  (z critical value)
p(1  p)
n
The desired confidence level determines the z critical
value. The three most commonly used confidence levels,
90%, 95%, and 99%, use z critical values 1.645, 1.96,
and 2.58, respectively.
9. The sample size required to estimate a population
proportion  to within an amount B with a confidence
level is
n = (1-) ( z critical value) 2
B
The value of  may be estimated using prior
information. In the absence of any such information,
using  = .5 in this formula gives a conservatively large
value for the required sample size.
10. The one-sample z confidence interval for 
When
1. x is the sample mean of a random sample,
2. the population distribution is normal or the sample
size n is large (generally n  30), and
3. the population standard deviation  is known
the formula for a confidence interval for a population
mean  is
x
 (z critical value) (  )
n
11. Let x1, x2, , xn be a random sample from a
normal population distribution. Then the probability
distribution of the standardized variable
t
x
s/
n
has the t distribution with n-1 df.
12. The one-sample t confidence interval for 
When
1. x is the sample mean of a random sample
2. the population distribution is normal or the sample
size n is large (generally n  30), and
3. the population standard deviation  is unknown
the formula for a confidence interval for population
mean  is
x
 (t critical value) (
s
n
)
where the t critical value is based on n-1 df, which
can be found by Appendix Table 3 on page 708.
13. The sample size required to estimate a population
mean  to within an amount B with a confidence level
is
n =[ ( z critical value) ]2 .
B
If  is unknown, it may be estimated based on
previous information or, for a population that is not
too skewed, by using (range)/4.
14. Important examples in the Notes:
Examples: 8.1, 8.2, 8.3, 8.4, 9.1, 9.2, 9.3, 9.4.
15. Exercise in class: A random sample of n = 12
four-year-old red pine trees was selected, and the
diameter (in inches) of each tree's main stem was
measured. The resulting observations are as follows:
11.3 10.7 12.4 15.2 10.1 12.1 16.2 10.5 11.4 11.0 10.7 12.0
(a) Give a point estimate of , the population mean
diameter.
(b) Give a point estimate of the population median
diameter.
(c) Give a point estimate of , the population
proportion of trees whose main stem diameters
are at least 12 inches.
(d) Compute a point estimate of σ, the population
standard deviation of main stem diameter.
(e) Suppose that the diameter distribution is normal.
Then the 90th percentile of the diameter
distribution is μ+1.28σ (so 90% of all trees have
diameters less than this value). Compute a point
estimate for this percentile.