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

Point Estimate for q (Population Proportion of Failures)

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. 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

A c Confidence Interval for p for Large Samples (np > 5 and nq > 5) z c = 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?

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?

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. so the use of the normal distribution is justified.

Out of 500 departures, 370 departed on time. Find a 99% confidence interval.

99% confidence interval for the proportion of airplanes that depart on time: E = Confidence interval is:

99% confidence interval for the proportion of airplanes that depart on time Confidence interval is < p < 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

A 95% confidence interval for population proportion p is:

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.

“... 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.