CH 6: Random Variables AP Statistics 12

6.3 Part II Normal Approximation for Binomial Distributions

Ex 1: At a certain factory that makes computer chips, the probability of obtaining a

defective chip at random is 0.21.

a) If 10 chips are drawn at random, sketch the distribution of X = the number of defective chips. After you sketch the distribution, sketch a smooth curve on the graph which models the distribution as closely as possible. (A graphing window of [-1,20] and [-0.1,0.35] works well)

b) Repeat the process for (a) using 30 chips. This time only sketch the smooth curve on the graph. (Keep the same window).

c) Repeat the process now using 50 chips. Again, only sketch the smooth curve and

keep the same window)

d) What do you notice as we increase the number of trials?

Hint: In L1 list #s 0 to 10 Now store: binomialpdf(10, 0.21, L1) as L2

Now Graph

CH 6: Random Variables AP Statistics 12

e) Now find the mean and standard deviation of the distribution from (c).

f) Find )10( XP for the distribution from (c) in two ways. First, use Binomcdf, then

Secondly, use Normalcdf(Low bound, Upper bound, , ). Compare your results.

What do you notice? Important Note: Because the binomial distribution is discrete and the normal distribution is continuous, we have to make an adjustment to our probability statements. This is called a continuity correction. The continuity correction means that you need to add an extra 0.5 to the ends of your lower or upper bounds. So, )10( XP in

the binomial setting becomes )5.10( XP in the normal setting.

g) Now find )10( XP (from (f)) using the continuity correction.

h) Find )12( XP , )83( XP , and )10( XP

Rule of when to use Normal Approx: We will use normal approximations to approximate binomial probabilities when:

10np and 10)1( pn

CH 6: Random Variables AP Statistics 12

6.3 Part II Assignment AP Statistics 12 1. When Bruce asks a girl to dance with him, his success rate is 30%. Suppose in any given night, he asks 20 girls to dance.

a) Verify that this is a binomial setting. Define the random variable X.

b) What is the likelihood that Bruce succeeds 10 times?

c) What is the likelihood that Bruce succeeds at least 10 times?

d) Find the mean and standard deviation of X.

e) Use a normal approximation to find the probability that Bruce will succeed at least 10 times. DON’T FORGET THE CONTINUITY CORRECTION!

f) Use a normal approximation to find the probability that Bruce will succeed at most 5 times.

g) Use a normal approximation to find the probability that Bruce will succeed between 10 and 15 times.

CH 6: Random Variables AP Statistics 12

2. Alina decides to wear a TEAL dress to her Grad Dinner/Dance. There is a 21% chance that another lady also wears TEAL. Suppose there are 40 ladies (other than Alina) going to Grad.

a) Verify that this is a binomial setting. Define the random variable T (for teal).

b) What is the chance that 11 other girls (Disastrous) are wearing teal?

c) What is the likelihood that at least 5 other girls are wearing teal?

d) Find the mean and standard deviation of T.

e) Use a normal approximation to find the probability that Alina will be wearing teal with at least 7 other girls. DON’T FORGET THE CONTINUITY CORRECTION!

f) Use a normal approximation to find the probability that Alina will wear teal with at most 5 other girls.

g) Use a normal approximation to find the probability that Alina will share teal with between 5 and 15 other girls.