Normal Distribution as an Approximation to the Binomial Distribution

Normal Distribution as an

Approximation

to the Binomial Distribution

2 : Goals ♦ ONE ♦ TWO ♦ THREE

Review

Review

Binomial Probability Distribution

v applies to a discrete random variable v has these requirements:

1. The experiment must have fixed number of trials.

2. The trials must be independent.

3. Each trial must have all outcomes classified into

two categories.

4. The probabilities must remain constant for each

trial.

4 The Normal Approximation to the

Binomial

♦ Using the normal distribution (a continuous distribution) as a substitute for a binomial distribution (a discrete distribution) for large values of n seems reasonable because as n increases, a binomial distribution gets closer and closer to a normal distribution.

♦ The normal probability distribution is generally deemed a good approximation to the binomial probability distribution when n and n(1- ) are both greater than 5.

π π

5 The Normal Approximation continued

♦ Recall for the binomial experiment:

nThere are only two mutually exclusive

outcomes (success or failure) on each trial.

nA binomial distribution results from

counting the number of successes.

nEach trial is independent.

nThe probability is fixed from trial to trial, and the number of trials n is also fixed.

Approximate a Binomial Distribution Approximate a Binomial Distribution

with a Normal Distribution if: with a Normal Distribution if: 1. np ≥ 5

Approximate a Binomial Distribution Approximate a Binomial Distribution

with a Normal Distribution if: with a Normal Distribution if: 1. np ≥ 5

2. nq ≥ 5

distribution.

(normal)

Then µ = np and σσ = npq and the random variable has a

8

Binomial Distribution for an n of 3 and 20, where π =.50 n=3 0 0.1 0.2 0.3 0.4 0 1 2 3 number of occurences P(x) n=20 0 0.05 0.1 0.15 0.2 2 4 6 8 10 12 14 16 18 20 n u m b e r o f o c c u r e n c e s P(x)

Solving Binomial Probability Problems Using a Normal Solving Binomial Probability Problems Using a Normal

Approximation Approximation

Solving Binomial Probability Problems Using a Solving Binomial Probability Problems Using a

Normal Approximation Normal Approximation

After verifying that we have a binomial probability problem, identify n, p, q

Is Computer Software

Available ?

Can the problem be solved by using Table A-1

?

Can the problem be easily solved

with the binomial probability formula

?

Use the Computer Software

Use the Table A-1

Use binomial probability formula

Yes Yes Yes No No Start P(x) = • p(n – x)!x! x • q n! 1 2 3 4 n–x

Solving Binomial Probability Problems Using a Solving Binomial Probability Problems Using a

Normal Approximation Normal Approximation

Can the problem be easily solved with the binomial probability formula?

Use binomial probability formula Yes P(x) = • p(n – x)!x! x • q n! 76 5 4 Are np ≥ 5 and nq ≥ 5 both true ? No No Yes Compute µ = np and σ =√ npq

Draw the normal curve, and identify the region representing the probability to be found. Be sure to include the continuity correction. (Remember, the discrete value x is adjusted for continuity by adding and subtracting 0.5)

n–x

Solving Binomial Probability Problems Using a Solving Binomial Probability Problems Using a

Normal Approximation Normal Approximation

8 9 7

Draw the normal curve, and identify the region representing the probability to be found. Be sure to include the continuity correction. (Remember, the discrete value x is adjusted for continuity by adding and subtracting 0.5)

Calculate

where µ and σ are the values already found and x is adjusted for continuity.

z =x – µ_σ

Refer to Table A-2 to find the area between µ and the value of x adjusted for continuity. Use that area

13

Normal Approximation to the Binomial: Continuity Correction Factor

♦ The value .5 subtracted or added, depending on the problem, to a selected value when a binomial probability distribution (a discrete probability distribution) is being approximated by a continuous probability distribution (the normal distribution).

Continuity Corrections Procedures Continuity Corrections Procedures 1. When using the normal distribution as an approximation to the

binomial distribution, always use the continuity correction. 2. In using the continuity correction, first identify the discrete whole

number x that is relevant to the binomial probability problem.

3. Draw a normal distribution centered about µ, then draw a vertical strip area centered over x . Mark the left side of the strip with the

number x − 0.5, and mark the right side with x + 0.5. For x = 64, draw a strip from 63.5 to 64.5. Consider the area of the strip to represent the probability of discrete number x.

continued

Continuity Corrections Procedures Continuity Corrections Procedures 4. Now determine whether the value of x itself should be included in the

probability you want. Next, determine whether you want the probability of at least x, at most x, more than x, fewer than x, or exactly x. Shade the area to the right of left of the strip, as appropriate; also shade the interior of the strip itself if and only if x

itself is to be included, The total shaded region corresponds to probability being sought.

x

x

x

x

x

x

x

x

x

x

x

exactly

x

exactly

Interval represents discrete number 64 64 50 64.5 63.5 50

x

exactly

Normal Approximation to the Binomial Example: Video camera

♦ A recent study by a marketing research firm showed that 15% of Canadian households owned a video camera. A sample of 200 homes is obtained.

♦ Of the 200 homes sampled how many would you expect to have video cameras?

µ

=

n

π

=

(.

15 200

)(

)

=

30

Normal Approximation to the Binomial Example: Video camera

♦ What is the variance?

♦ What is the standard deviation?

♦ What is the probability that less than 40 homes in the sample have video cameras? We need P(X<40)=P(X<39).

♦ So, using the normal approximation, P(X<39.5)

σ 2 π π 1 3 0 1 1 5 2 5 5 = n ( − ) = ( ) ( −. ) = . σ = 2 5 5. = 5 0 4 9 8. ≈ ≈

25 - 5 0. 4 0. 3 0. 2 0. 1 . 0 f ( x r al i t rbu i on: µ = 0, σ2 = 1

Normal Approximation to the Binomial Example: Video camera 0 1 2 3 4 P(Z=1.88) = 0.5+ 0.4699 = 0.9699 Z=1.88 26 Summary • •