Math 3680 Lecture #10 Normal Random Variables Lecture #11

Accordingly, about 68% of women have heights Many distributions follow the bell curve rather well. For example, according to studies, the height of U.S. women has the following data: m = 63.5 inches and s = 2.5 inches Accordingly, about 68% of women have heights between 61 and 66 inches.

However, the bell curve does not model all data sets well.

Example: What percentage of women has heights between 60 and 68 inches?

Example: What percentage of women has heights between 60 and 68 inches? Is this answer exact or an approximation?

Example: For a certain population of high school students, the SAT-M scores are normally distributed with m = 500 and s = 100. A certain engineering college will accept only high school seniors with SAT-M scores in the top 5%. What is the minimum SAT-M score for this program? Note: Before, the kind of question that was posed was: “What percentage of students score above 700 on the SAT-M?” Now, the percentage is specified.

Approximating a Binomial(n, p) distribution with a normal curve

Example: From a heterozygous cross of two pea plants, 192 seeds are planted. According to the laws of genetics, there is a 25% chance that any one offspring will be recessive homozygous, independently of all other offspring. Find the probability that between 44 and 55 (inclusive) of the seeds are recessive homozygous.

Solution #1 (Exact). Let X denote the number of recessive homozygous offspring. Then X ~ Binomial(192, 0.25) Therefore,

In terms of the probability histogram:

Solution #2 (Approximate) Solution #2 (Approximate). Note that the probability histogram of the Binomial(192, 0.25) distribution is approximated well by a bell curve. Since X ~ Binomial(192, 0.25), we have We convert 43.5 and 55.5 to standard units:

Solution #2 (continued) Therefore, Question: Why did we standardize 43.5 and 55.5? This subtlety is called continuity correction.

BBinomial(n, 0.5) distribution

BBinomial(n, 0.5) distribution

BBinomial(100, p) distribution

BBinomial(100, p) distribution

BBinomial(n,π) distribution with π small

Limitations On Normal Approximation The normal approximation is reasonable if n is large and both p and 1 - p are not small. More precisely, the approximation is good if n p  5 and n (1 - p)  5. What happens if n is large and p is small? What if instead p is close to 1?

Continuity Correction The primary difficulty with using the continuity correction is deciding whether to add or subtract 0.5 from the endpoints. This decision may be facilitated by drawing a rough histogram and then deciding which rectangles are to be included. Example: In the previous problem, what should be converted to standard units to find P(24  X  30) P(31 < X  38) P(X > 25) P(X  37)

Continuity Correction There’s nothing particularly special about the binomial distribution for this procedure. The continuity correction can be used whenever a discrete random variable is being approximated by a continuous one.

Example: A gambler repeatedly bets on red in roulette Example: A gambler repeatedly bets on red in roulette. The chance of winning on one play is p = 9/19. Suppose the gambler plays 100 times. Use the normal approximation to estimate the probability that the gambler wins more than 50 times.