Normal Probability

Distributions

Intro to Normal Distributions & the STANDARD Normal Distribution

1. mean, median, and mode are equal

2. bell shaped and symmetric about the mean

3. total area under the curve is 1

4. the curve approaches, but never touches the x axis as it extends away from the mean

A Normal distribution with mean = 0 and standard deviation = 1

The cumulative area is close to 0 for z-scores close to z = -3.49

The cumulative area increases as the z-score increases.

The cumulative area for z = 0 is 0.5000

The cumulative area is close to 1 for z-scores close to z = 3.49

1. Sketch the curve and shade the appropriate area under the curve.

2. To find the area LEFT of z, find the area that corresponds to z on the table.

3. To find the area RIGHT of z, find the area that corresponds to z on the table, then subtract from 1.

4. To find the area BETWEEN 2 z-scores, find each area, then subtract.

1. left of z = -1.02 2. right of z = 2.30 3. left of z = 0.45 4. right of z = - 1.99 5. between z = -2.34 and z

= 1.89

Normal Distributions: Finding Probabilities

Find P(x < 200)

Find P(x > 155)

Find P(172 < x < 192)

In a survey of US men, the heights in the 20 – 29 age group were normally distributed, with a mean of 69.4 inches and a standard deviation of 2.9 inches. Find the probability that a randomly selected participant has a height that is

A) less than 66 in. B) between 66 and 72 in. C) more than 72 in. D) Identify any unusual events.

Explain your reasoning.

The amounts of time per workout an athlete uses a stairclimber are normally distributed, with a mean of 20 minutes and a standard deviation of 5 minutes. Find the probability that a randomly selected athlete uses a stairclimber for

A) less than 17 minutes. B) between 20 and 28 minutes. C) more than 30 minutes.

SAT math scores are normally distributed with a mean of 514 and a standard deviation of 117.

A) What percent of the SAT math scores are less than 500?

B) Out of 1500 randomly selected SAT math scores, about how many would you expect to be greater than 600?