Math 10 Chapter 6 Notes: The Normal Distribution

Notation: X is a continuous random variable X ~ N(, )

Parameters: is the mean and is the standard deviation

Graph is bell-shaped and symmetrical The mean, median, and mode are the same

(in theory)

Math 10 Chapter 6 Notes: The Normal Distribution

Total area under the curve is equal to 1. Probability = Area

P(X < x) is the cumulative distribution function or Area to the Left.

A change in the standard deviation, , causes the curve to become wider or narrower

A change in the mean, , causes the graph to shift

Math 10 Chapter 6 Notes: The Standard Normal Distribution

A normal (bell-shaped) distribution of standardized values called z-scores.

Notation: Z ~ N(0, 1) A z-score is measured in terms of the standard deviation. The formula for the z-score is

x

z

Math 10 Chapter 6 Notes: The Normal Distribution

Bell-shaped curve Most values cluster

about the mean Area within 4 standard

deviations (+ or - 4 ) is 1

Math 10 Chapter 6 Notes: The Normal Distribution

Ex. Suppose X ~ N(100, 5). Find the z-score (the standardized score) for x = 95 and for 110.

x

z = 95 – 100 = - 1

5

x

z= 110 – 100 = 2

5

Math 10 Chapter 6 Notes: The Normal Distribution

 The z-score lets us compare data that are scaled differently. Ex. X~N(5, 6) and Y~N(2, 1) with x = 17 and y = 4; X = Y = weight gain

17 – 5 = 2 4 – 2 = 2 6 1

Math 10 Chapter 6 Notes: The Standard Normal Distribution

 Ex. Suppose Z ~ N(0, 1). Draw pictures and find the following.

1.  P(-1.28 < Z < 1.28)

2.  P(Z < 1.645)

3.  P(Z > 1.645)

4.  The 90th percentile, k, for Z scores.

 For 1, 2, 3 use the normal cdf

For 4, use the inverse normal

Math 10 Chapter 6 Notes: The Normal Distribution

Ex: At the beginning of the term, the amount of time a student waits in line at the campus store is normally distributed with a mean of 5 minutes and a standard deviation of 2 minutes.

Let X = the amount of time, in minutes, that a student waits in line at the campus store at the beginning of the term.

X ~ N(5, 2) where the mean = 5 and the standard deviation = 2.

Math 10 Chapter 6 Notes: The Normal Distribution

Find the probability that one randomly chosen student waits more than 6 minutes in line at the campus store at the beginning of the term.

P(X > 6) = 0.3085.

Math 10 Chapter 6 Notes: The Normal Distribution

Find the 3rd quartile. The third quartile is equal to the 75th percentile.

Let k = the 75th percentile

(75th %ile).

P(X < k ) = 0.75.

The 3rd quartile or 75th percentile is 6.35 minutes (to 2 decimal places). Seventy-five percent of the waiting times are less than 6.35 minutes.