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Math 10 Chapter 6 Notes: The Normal Distribution

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- 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)

- 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

- 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

- Bell-shaped curve
- Most values cluster about the mean
- Area within 4 standard deviations (+ or - 4 ) is 1

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

= 95 – 100 = - 1

5

= 110 – 100 = 2

5

· 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

6 1

· 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

(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.