Download Binomial Experiments

Section 6.3:
Confidence Intervals for Population Proportions
Today we will study (Part II)
• One minute review of binomial experiments
• Point estimate for the Population Proportion p
• Construction of confidence intervals for a population proportion
Binomial Experiments
A binomial experiment is a probability experiment that satisfies the following conditions.
1. The experiment is repeated for a fixed number of trials.
2. Each trial is independent of the other trials (the out come of any individual trial doesn’t affect the probabilities in
the other trials).
3. There are only two possible outcomes of interest for each trial. The outcomes can be classified as either a success
(S) or as a failure (F).
4. The probability of a success P(S) is the same for each trial.
The random variable x counts the number of successful trials.
Notation, Binomial Experiments
Symbol
n
Description
The number of times a trial is repeated
p = P (S)
The probability of success in a single trial
q = P (F )
The probability of failure in a single trial (q = 1 − p)
x
The random variable represents a count of the number of successes in n trials.
Normal Approximation to a Binomial Distribution
If np ≥ 5 and nq ≥ 5, then the binomial random variable x is approximately normally distributed, with
µ = np
σ=
√
npq
mean
standard deviation
Paraphrased from our text - section 6.3
The requirements for the methods of this section:
1. The conditions (4) for the binomial experiment are satisfied.
2. The normal distribution can be used to approximate the distribution of sample proportions.
3. The sample is a simple random sample.
We are still trying to estimate a population parameter, this time the population proportion. The sample proportion p̂ is
the best point estimate of the population proportion p.
DEFINITION
The point estimate for p, the population proportion of successes, is given by the proportion of successes in a
sample and is denoted by
p̂ =
x
n
where x is the number of successes in the sample and n is the number in the sample.
The point estimate for the proportion of failures is q̂ = 1 − p̂.
The symbols p̂ and q̂ are read as “p hat” and “q hat”.
Confidence Intervals for a Population Proportion p
DEFINITION
A c-confidence interval for the population proportion p is
p̂ − E < p < p̂ + E
where the margin of error E is
r
p̂q̂
.
E = zc
n
The probability that the confidence interval contains p is c.
Often written in interval notation: (p̂ − E, p̂ + E).
GUIDELINES
Constructing a Confidence Interval for a Population Proportion
1. Identify the sample statistics n and x.
x
n
2. Find the point estimate p̂.
p̂ =
3. Verify that the sampling distribution of p̂
can be approximated by the normal distribution.
np̂ ≥ 5, nq̂ ≥ 5
4. Find the critical value zc that corresponds to the
given level of confidence c.
Use the Standard Normal Table
q
p̂q̂
n
5. Find the margin of error E.
E = zc
6. Find the left and right endpoints and
write down the confidence interval.
Interval:
p̂ − E < p < p̂ + E
Round-off Rule
Round the confidence interval endpoints to three decimal places.
STATISTICS - M109