Using Normal Distribution to Approximate a Discrete Distribution

Discrete Distributions Probability Distribution of a discrete variable is represented by a histogram

Discrete Distributions In some cases It can be approximated By a normal distribution

The number of express mail packages mailed at a given office is approximately normal with a mean of 18 and standard deviation of 6. A Distribution has Approximately Normal Distribution What is the probability that x=20? In a Normal Distribution, probability is represented by the area under a curve. So the probability that it is “exactly” 20 is zero.

The number of express mail packages mailed at a given office is approximately normal with a mean of 18 and standard deviation of 6. A Distribution has Approximately Normal Distribution What is the probability that x=20? In a Normal Distribution, probability is represented by the area under a curve. The probability that it is between 19.5 and 20.5 is.0641, or 6.41% Replace the whole number r With the range r to r +0.5 This is the continuity correction. Find the probability between and

An IQ Score must be a whole number, so is represented by a discrete distribution. Assume that the distribution can be approximated with a normal distribution with a mean of 100 and standard deviation of 15. IQ Scores Find P(x=100) (x=100) =>(99.5 ≤ x ≤ 100.5)

An IQ Score must be a whole number, so is represented by a discrete distribution. Assume that the distribution can be approximated with a normal distribution with a mean of 100 and standard deviation of 15. IQ Scores Find P(x ≤ 110) (x ≤ 110) =>(x ≤ 110.5) Find P(x < 110) (x (x ≤ 109) (x ≤ 109) =>(x ≤ 109.5)

An IQ Score must be a whole number, so is represented by a discrete distribution. Assume that the distribution can be approximated with a normal distribution with a mean of 100 and standard deviation of 15. IQ Scores Find P(75 ≤ x ≤ 125) (75 ≤ x ≤ 125) =>(74.5 ≤ x ≤ 125.5)

Premature Babies: Premature babies are born more than 3 weeks early. 10% of live births in US are premature and we have a sample of 250 births. Normal Approximation of Binomial Distribution The distribution can be approximated by a Normal Distribution, because np = 250(.10) = 25 and nq = 250(.9) = 225 are both > 5. Mean = np = 25 This is a binomial distribution because a baby either is premature or is not.

Premature Babies: Premature babies are born more than 3 weeks early. If 10% of live births in US are premature and we have a sample of 250 births : Normal Approximation of Binomial Distribution The distribution can be approximated by a Normal Distribution, With mean = 25 and standard deviation = What is the probability that the number of premature births in our sample will be between 15 and 30 (inclusive)? Find P(15 ≤ x ≤ 30) (15 ≤ x ≤ 30) =>(14.5 ≤ x ≤ 30.5)

Premature Babies: Normal Approximation of Binomial Distribution The distribution can be approximated by a Normal Distribution, With mean = 25 and standard deviation = What is the probability that the number of premature births in our sample will be between 15 and 30 (inclusive)? Find P(15 ≤ x ≤ 30) (15 ≤ x ≤ 30) =>(14.5 ≤ x ≤ 30.5) (-2.21 ≤ z ≤ 1.16)