Download Section 8.3 A confidence interval to estimate one population mean.

Section 8.3
A confidence interval to estimate
one population mean.
Review
A point estimate is a single number
that is our “best guess” for the
parameter.
◦  The sample proportion is a point
estimate of the population proportion.
◦  The sample mean is a point estimate of
the population mean.
Point Estimates
A good estimator has two properties:
◦  The sampling distribution for the estimator is
centered at the parameter of interest.
–  Note that this is true for both the sampling
distribution of the mean and proportion.
◦  The sampling distribution for the estimator
has a relatively small standard error.
–  Again, this is true for both mean & prop.
Review
An interval estimate is an interval
of numbers within which the
parameter is believed to fall.
◦  Because interval estimates contain the
parameter with a certain degree of
confidence these are often called
confidence intervals.
Confidence Intervals
—  We
build the confidence interval
from the sampling distribution of
the point estimator.
—  We
need an appropriately large
sample size.
◦  For proportions: np > 15
& n(1 - p) > 15
◦  For means: n > 30
Confidence Intervals
—  The
larger the confidence level, the
wider the interval will be.
—  The
larger the sample size, the
______ the standard error will be.
RECALL: Confidence Intervals for
Proportions
— 
Use the sample proportion to estimate the
standard error of the sampling distribution:
s.e. ≈
pˆ (1 − pˆ )
n
Determine how many standard errors to
use by looking up the confidence level in
the normal table to find the associated z.
—  The confidence interval is then:
— 
( pˆ − z * s.e., pˆ + z * s.e.)
NEW: Confidence Intervals for
Means
Estimate a population mean µ
using a confidence interval of the
usual form:
point estimate ± margin of error
NEW: Confidence Intervals for
Means
— 
Use the sample standard deviation to
estimate the standard error of the sampling
s
distribution: s.e. ≈
n
Because we are approximating the mean
and the standard deviation in this case, the
distribution is bell-shaped but not exactly
normal. It actually has a t-distribution.
—  The t-distribution has a bell-shape, gets
more normal as n heads to infinity and has
thicker tails than a normal distribution.
— 
Confidence Intervals for Means
Determine how many standard errors to
use by looking up the confidence level in
the t-table to find the associated t.
—  The confidence interval is then:
— 
(x − t * s.e., x + t * s.e.)
The t*s.e. part is the margin of error.
—  If we happen to know the value of σ we can
use the normal distribution.
—  If the population distribution is not normal
the t-distribution is still robust. It
performs adequately well in this case.
— 
Example
Use a 95% CI to estimate the
average number of hours per week
that SU students work.
◦  Population?
◦  Categorical or quantitative variable?
◦  Parameter of interest?
Example
Suppose we randomly sample 40 SU students
and find that the average number of hours that
they work per week is 4.2 with a standard
deviation of 2.8.
—  Symbols:
— 
Point estimate:
Standard error:
Margin of error:
— 
CI Interval with Table B & with TI calculator.
— 
— 
Confidence Intervals for Means
Suppose that an experiment determines a
95% confidence interval for weight of a
particular candy bar (in grams) of
(8.1, 9.2).
—  This means that we are 95% confident that
the actual average weight of these candy
bars is between 8.1 and 9.2 grams.
—  Approximately 5% of the time a sample of
candy bars will produce a confidence
interval that doesn’t contain the true
average weight.
—