Transcript Review of the Binomial Distribution

Review of the Binomial
Distribution
• Completely determined by the number of
trials (n) and the probability of success (p)
in a single trial.
• q=1–p
• If np and nq are both > 5, the binomial
distribution can be approximated by the
normal distribution.
A Point Estimate for p, the
Population Proportion of
Successes
r
pˆ ( read as " p hat " ) 
n
Point Estimate for q
(Population Proportion of
Failures)
qˆ ( read as " q hat " )  1  pˆ
For a sample of 500 airplane
departures, 370 departed on
time. Use this information to
estimate the probability that an
airplane from the entire
population departs on time.
r 370
pˆ  
 0.74
n 500
We estimate that there is a 74% chance that any
given flight will depart on time.
Error of Estimate for “p hat” as
a Point Estimate for p
pˆ  p
A c Confidence Interval for p
for Large Samples
(np > 5 and nq > 5)
pˆ  E  p
 pˆ  E
r
where pˆ 
and E  z
n
c
pˆ (1  pˆ )
n
zc = critical value for confidence level c
taken from a normal distribution
For a sample of 500 airplane
departures, 370 departed on
time. Find a 99% confidence
interval for the proportion of
airplanes that depart on time.
Is the use of the normal distribution
justified?
n  500
pˆ  0.74
For a sample of 500 airplane
departures, 370 departed on
time. Find a 99% confidence
interval for the proportion of
airplanes that depart on time.
Can we use the normal distribution?
npˆ  370
nqˆ  130
For a sample of 500 airplane
departures, 370 departed on
time. Find a 99% confidence
interval for the proportion of
airplanes that depart on time.
npˆ and
nqˆ are both  5
so the use of the normal distribution is justified.
Out of 500 departures, 370
departed on time. Find a 99%
confidence interval.
r 370
pˆ  
 0.74
n 500
.74(.26)
E  2.58
 0.0506
500
99% confidence interval for the
proportion of airplanes that
depart on time:
E = 0.0506
Confidence interval is:
ˆ E  p p
ˆE
p
.74  0.0506  p  .74  0.0506
0.6894  p  0.7906
99% confidence interval for
the proportion of airplanes that
depart on time
Confidence interval is
0.6894 < p < 0.7906
We are 99% confident that between 69%
and 79% of the planes depart on time.
The point estimate and the
confidence interval do not
depend on the size of the
population.
The sample size, however, does
affect the accuracy of the
statistical estimate.
Margin of Error
The margin of error is the maximal
error of estimate E for a
confidence interval.
Usually, a 95% confidence interval
is assumed.
Interpretation of Poll Results
The proportion responding in a certain way is
pˆ
A 95% confidence interval for
population proportion p is:
pˆ  m arg in of error  p  pˆ  m arg in of error
pˆ  poll report
Interpret the following poll
results:
“ A recent survey of 400 households
indicated that 84% of the households
surveyed preferred a new breakfast cereal
to their previous brand. Chances are 19
out of 20 that if all households had been
surveyed, the results would differ by no
more than 3.5 percentage points in either
direction.”
“Chances are 19 out of 20 …”
19/20 = 0.95
A 95% confidence interval is being
used.
“... 84% of the households
surveyed preferred …”
84% represents the percentage of
households who preferred the
new cereal.
84% represents pˆ .
“... the results would differ by
no more than 3.5 percentage
points in either direction.”
3.5% represents the margin of
error, E.
The confidence interval is:
84% - 3.5% < p < 84% + 3.5%
80.5% < p < 87.5%
The poll indicates ( with 95%
confidence):
between 80.5% and 87.5% of the
population prefer the new
cereal.