1. Use the pivotal
variable Z to show that an approximate 1 - r level confidence interval,
confidence upper bound, and confidence lower bound for p are given as follows:
Virtual Laboratories > Interval Estimation > 1 2 3 [4] 5 6
Suppose that I1, I2, ..., In is a random sample from the Bernoulli distribution with unknown parameter p in (0, 1). Thus, these are independent random variables taking the values 1 and 0 with probabilities p and 1 - p respectively. Usually, this model arises in one of the following contexts:
In this section, we will construct confidence intervals for p. A parallel section on Tests in the Bernoulli Model is in the chapter on Hypothesis Testing.
Recall that the mean and variance of the Bernoulli distribution are
E(I) = p, var(I) = p(1 - p).
Note that the sample mean M is the sample proportion of objects of the type of interest. By the central limit theorem,
Z = (M - p) / [M(1 - M) / n]1/2
has approximately a standard normal distribution and hence is (approximately) a pivot variable for p.
1. Use the pivotal
variable Z to show that an approximate 1 - r level confidence interval,
confidence upper bound, and confidence lower bound for p are given as follows:
The distribution of Z is closest to normal when p is near 1/2 and farthest from normal when p is near 0 or 1 (extreme).
2. Use the
simulation of the proportion estimation experiment to explore the procedure. Use various
values of p and various confidence levels, sample sizes, and interval types. For
each configuration, run the experiment 1000 times with an update frequency of 10 and note
how well the proportion of successful intervals approximates the theoretical confidence
level.
3. Show that the
variance of the Bernoulli distribution is maximized when p = 1/2 and thus the
maximum variance is 1/4.
4. Use the result
of the previous exercise to show that a conservative 1 - r level
two-sided confidence interval, confidence lower bound, and confidence upper bound for p
are given as follows:
Thus, the conservative confidence intervals will be larger than the confidence intervals using the first procedure. The conservative estimate can be used to design the experiment.
5. Suppose that p
is to be estimated with margin of error E and with 1 - r confidence.
Show that a conservative estimate of the sample size is
n = ceil[(z / 2E)2]
where z = z1 - r/2 for a two-sided interval and z = z1 - r for a one-sided confidence interval.
6. In a pole of
1000 registered voters in a certain district, 427 prefer candidate X. Construct the 95%
two-sided confidence interval for the proportion of all registered voters in the district
that prefer X.
7. A coin is
tossed 500 times and results in 302 heads. Construct the 95% confidence lower bound for
the probability of heads. Do you believe that the coin is fair?
8. A sample of 400
memory chips from a production line are tested, and 30 are defective. Construct the
conservative 90% two-sided confidence interval for the proportion of defective chips.
9. A drug company
wants to estimate the proportion of persons who will experience an adverse reaction to a
certain new drug. The company wants a two-sided interval with margin of error 0.03 with
95% confidence. How large should the sample be?
10. An advertising
agency wants to construct a 99% confidence lower bound for the proportion of dentists who
recommend a certain brand of toothpaste. The margin of error is to be 0.02. How large
should the sample be?