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).
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Ex. Suppose X ~ N(100, 5). Find the z-score (the standardized score) for x = 95 and for 110.
= 95 – 100 = - 1
= 110 – 100 = 2
· 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 = 24 – 2 = 2
· 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
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.
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.
Find the 3rd quartile. The third quartile is equal to the 75th percentile.
Let k = the 75th percentile
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.