Download 5.5 Normal Approximation to Binomial Distributions

Statistics
5.5 Normal Approximations to Binomial Distributions
Objectives: Approximating a Binomial Distribution; Correction for Continuity; Approximating
Binomial Probabilities
Procedure:
1. Approximating a Binomial Distribution:
a. Normal Approximation to a Binomial Distribution:
If np ≥ 5 and nq ≥ 5, then the binomial random variable x is approximately
normally distributed, with mean
And standard deviation
b. Example 1: Approximating the binomial distribution:
Two binomial experiments are listed. Decide whether you can use the normal
distribution to approximate x, the number of people who reply yes. If you can, find
the mean and standard deviation. If you cannot, explain why.
a. 34% of people in the US say that they are likely to make a New Year’s
resolution. You randomly select 15 people in the US and ask each if he
or she is likely to make a New Year’s resolution.
b. 6% of people in the US who made a New Year’s resolution resolved to
exercise more. You randomly select 65 people in the US who made a
resolution and ask each if he or she resolved to exercise more.
2. Correction for Continuity:
a. Example 2: Using a correction for continuity:
Use a correction for continuity to convert each of the following binomial intervals
to a normal distribution interval.
a. The probability of getting between 270 and 310 successes, inclusive.
b. The probability of at least 158 successes.
c. The probability of getting less than 63 successes.
d. The probability of getting between 57 and 83 success, inclusive.
e. The probability of getting at most 54 successes.
3. Approximating Binomial Probabilities:
a. Guidelines: Using the normal distribution to approximate binomial probabilities:
i. Verify that the binomial distribution applies.
ii. Determine if you can use the normal distribution to approximate x, the
binomial variable.
iii. Find the mean and standard deviation for the distribution.
iv. Apply the appropriate continuity correction. Shade the corresponding area
under the curve.
v. Find the corresponding z-score(s).
vi. Find the probability.
b. Example 3: Approximating a binomial probability:
Thirty-four percent of people in the U.S. say that they are likely to make a New
Year’s resolution. You randomly select 15 people in the U.S. and ask each if he
or she is likely to make a New Year’s resolution. What is the probability that fewer
than eight of them respond yes?
c. Example 4: Approximating a binomial probability:
Thirty-eight percent of people in the U.S. admit that they snoop in other people
medicine cabinets. You randomly select 200 people in the U.S. and ask each if he
or she snoops in other people’s medicine cabinets. What is the probability that at
least 70 will say yes?
What is the probability that at most 85 people will say yes?
d. Example 5: Approximating a binomial probability:
A survey reports that 95% of Internet users use Microsoft Internet Explorer as
their browser. You randomly select 200 Internet users and ask each whether he
or she uses Microsoft Internet Explorer as his or her browser. What is the
probability that exactly 194 will say yes?
What is the probability that exactly 191 people will say yes?
4. HW: p. 265 (2 – 26 evens)