CHAPTER 2: DESCRIBING LOCATION IN A DISTRIBUTIONSection 2- Normal Distributions

NORMAL DENSITY CURVE

WHY NORMALS????

1. Offer good descriptions for some distributions of real data.

Examples: Standardized test scores Repeated careful measurements of the same quantity Characteristics of biological populations

2. Good approximation to results of many kinds of chance outcomes

3. Many statistical inference procedures based on Normal distn’s work well for other roughly symmetric distn’s

68-95-99.7 RULE

In the Normal distribution with mean µ and standard deviation σ

Appx 68% of the observations fall within σ of the mean µ

Appx 95% of the observations fall within 2σ of the mean µ

Appx 99.7% of the observations fall within 3σ of the mean µ

c

TRY IT OUT……

The distribution heights of young women aged 18-24 is approximately Normal with mean µ=64.5 inches, and standard deviation σ=2.5 inches. Write the graph to depict the 68-95-99.7 rule.

NORMAL DISTRIBUTION NOTATION

Using mean µ and standard deviation σ we abbreviate Normal distributions as:

N(µ, σ)

How could we abbreviate the last example in Normal distn notation?

STANDARD NORMAL DISTRIBUTION

The standard Normal distribution has mean 0 and standard deviation 1.

A standard Normal distribution is the set of all z-scores.

ANALYZE IT!!!

Suppose SAT scores among college students are normally distributed with a mean of 500 and a standard deviation of 100. If a student scores a 700, what would be her z-score?

ANALYZE IT!!!

• A set of math test scores has a mean of 70 and a standard deviation of 8.

• A set of English test scores has a mean of 74 and a standard deviation of 16.

For which test would a score of 78 have a higher standing?

ANALYZE IT!!!

What will be the miles per gallon for a Toyota Camry when the average mpg is 23, it has a z value of 1.5 and a standard deviation of 5?

AREA UNDER THE CURVE: Z-SCORE

Finding proportions of observation is from the standard Normal distribution given a z-score.

FINDING AREA GIVEN Z-SCORE

Find the area under the curve that lies to the left of z=1.46.

FINDING AREA GIVEN Z-SCORE

Find the area under the curve that lies to the left of z=1.46.

FINDING AREA GIVEN Z-SCORE

Find the area under the curve that lies to the right of z=1.46.

FIND THE P-VALUES…

Find the p-value for the area that lies to the left of z=-0.58.

Find the p-value for the area that lies between z=-1.16 and z=2.71.

MORE P-VALUES

Find the z-score given the p-value of 0.905.

Find the z-score given the p-value of 0.0735.