Download 5.4 Normal Approximation of the Binomial Distribution

5.4 Normal Approximation of
the Binomial Distribution
Review from yesterday….
Bernoulli Trials have 3 properties:
1. Only two outcomes - PASS or FAIL
2. n identical trials
3. Trials are independent - probability is
constant
Binomial Probability Distribution
In a binomial experiment with n Bernoulli
trials, each with a probability of success p, the
probability of k successes in the n trials is
given by:
Where X is the discrete random variable
corresponding to the number of successes.
Warm-Up
a) What is the probability of getting exactly 30 heads if a coin is
tossed 50 times?
X = # of heads in 50 tosses
n = # of Bernoulli Trials (50)
p = probability of success (.5)
b) Determine the expected number of heads in 50 trials
For a binomial distribution, expected value can simply be calculated by:
E(X) = np
Note:
n = # of trials
p = probability of success
c) Find the probability of tossing between 20 and 30 heads,
inclusive, in 50 tosses of a coin. P(20 ≤ X ≤ 30)
This is where the real problem occurs with more complex
situations. You could calculate it with 11 such calculations:
P(20 ≤ X ≤ 30) = P(X = 20) + P(X = 21) + P (X = 22) + … P(X = 30)
This calculation would be very time consuming! To help simplify
the calculation, look at the graphical representation of the
binomial distribution. A binomial distribution can be
approximated by a normal distribution as long as the number of
trials is large!
- It is mound shaped and the normal curve is a close fit.
- To use the normal curve, you must know values for the mean and
standard deviation because you must calculate the z-score in order
to translate the data from the normal curve to the standard normal
curve. This will allow you to determine the percentage of data that
has an equal or lower z-score. The percentage is equivalent to the
area under the curve.
What is the Standard Normal Curve?
The standard normal distribution is a special normal
distribution with a mean of 0 and a standard deviation of 1.
-3
-2
-1
0
1
2
3
Mean and Standard Deviation of the Normal
Approximation to the Binomial Distribution
Mean of a Binomial Distribution:
(** same as expected value)
Standard Deviation:
Practice:
If you were to toss a coin 50 times, calculate….
a) the mean ( x ) number of heads in 50 tosses
b) the standard deviation ( 𝛔 ) of the number of heads in 50 tosses
From Discrete to Continuous
There is one major issue…..
- A binomial distribution represents a discrete random variable
- A normal distribution is continuous
- In order to use the normal distribution to approximate the
binomial distribution, you must consider a range of values
rather than specific discrete values.
Example: To find a range of values to represent the discrete
value of 5, you must include all numbers that round to 5.
Therefore, the range of values between 4.5 and 5.5 can be used
to represent the discrete value of 5.
P(X = 5) = P(4.5 < X < 5.5)
Practice finding the range of continuous values for a
discrete value….
a) What range of continuous
values can be used to represent
P(X = 2) ?
b) What range of continuous
values can be used to represent
P(2 ≤ X ≤ 4) ?
c) What range of continuous
values can be used to represent
P(X > 3) ?
d) What range of continuous
values can be used to represent
P(X ≥ 3) ?
e) What range of continuous
values can be used to represent
P(2 < X < 5) ?
f) What range of continuous
values can be used to represent
P(X < 11) ?
g) What range of continuous
values can be used to represent
P(X ≥ 7) ?
Example 1
Fran tosses a fair coin 50 times. Estimate the probability that she
will get tails less than 20 times.
This means you must find P(X < 20) where X is the number of
tails in 50 tosses)
Step 1: Find the mean and standard deviation
Note: Let a success
be a toss of tails
Step 2: Determine continuous values for P(X < 20)
Step 3: Find the z-score
Note:
finding the z-score
standardizes the
distribution to the
standard normal curve
Step 4: Use your z-score to determine the probability using
the chart [Find P(z < -1.55)]
P(X < 19.5) = P(z < -1.55)
=
Therefore there is a _____ % chance that Fran will toss less
than 20 tails in 50 attempts.
Example 2
Calculate the probability that, in 100 rolls of a fair die, a 6 appears
between 10 and 20 times, inclusive. [Find P(10 ≤ X ≤ 20)]
Step 1: Find the mean and standard deviation
Step 2: Find the continuous range for the discrete values
of P(10 ≤ X ≤ 20)
Step 3: Find the z-scores
Step 4: Use your z-scores to determine the probability using
the chart
-3
-2
-1
0
1
2
3
P(9.5 < X < 20.5) = P(-1.92 < z <1.03)
= P(z < 1.03) - P(z < -1.92)
Therefore the probability of rolling between 10 and 20 (inclusive)
sixes on a fair die that is rolled 100 times is ___________%.
Using TI-83
Example 2 using your calculator
A TI-83 calculator can be used to calculate the area under the
normal distribution.
Use the command:
normalcdf(lower X value, upper X value, mean, stand. dev.)
access the command by pressing: 2nd --> distr --> 2
Example 3
A hamburger patty producer claims that its burgers contain 400
grams of beef. It has been determined that 85% of burgers contain
400 grams or more. An inspector will only accept a shipment if at
least 90% of a sample of 250 burgers contain more than 400 grams.
What is the probability that a shipment is accepted?
Step 1: Find the mean and standard deviation
Step 2: Find the continuous range for the discrete values
of P(X ≥ 225)
Note: we are using
225 because 90% of
250 burgers is 225.
Step 3: Find the z-score
Step 4: Use your z-score to determine the probability using
the chart
P(X > 224.5) = P(z > 2.12)
= 1 - P(z < 2.12)
=
There is a ______% chance that the shipment is accepted.
WARNING
Not all binomial distributions can be
approximated using normal distributions.
If X is a binomial random variable with n independent
trials with a probability of success = p, AND if
np > 5
and n(1-p) > 5
then you can approximate it using normal distribution
with mean=np and S.D.= np(1-p)
Was a normal approximation of the binomial distribution
appropriate for example 3?
np =
n(1 - p) =
Since np > 5 and 1(n - p) > 5, the binomial distribution can be
approximated by the normal curve.