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.
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.
74.0500
370ˆ n
rp
We estimate that there is a 74% chance that any given flight will depart on time.
A c Confidence Interval for p for Large Samples (np > 5 and nq > 5)
zc = critical value for confidence level c taken from a normal
distribution
npp
zEandnr
pwhere
EppEp
c
)ˆ1(ˆˆ
ˆˆ
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?
74.0ˆ500 pn
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?
130ˆ370ˆ qnpn
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.
5ˆˆ bothareqnandpn
so the use of the normal distribution is justified.
Out of 500 departures, 370 departed on time. Find a 99%
confidence interval.
74.0500
370ˆ n
rp
0506.0500
)26(.74.58.2 E
99% confidence interval for the proportion of airplanes that
depart on time:
E = 0.0506
Confidence interval is:
7906.06894.0
0506.074.0506.074.
ˆˆ
p
p
EppEp
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.
A 95% confidence interval for population proportion p is:
reportpollp
errorofinmpperrorofinmp
ˆ
argˆargˆ
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.”
“... 84% of the households surveyed preferred …”
84% represents the percentage of households who preferred the
new cereal. .ˆ represents %84 p
“... 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%
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