6.3 AREA UNDER ANY NORMAL CURVE

Chapter 6:

Normal Curves and Sampling Distributions

Normal Distribution Areas

In many applied situations, the original normal curve is not the standard normal curve.

How to Work with Normal Distributions To find areas and probabilities for a random

variable x that follows a normal distribution with a mean μ and standard deviation σ, convert x values to z values using the formula

Then use Table 5 (or calculator) to find corresponding areas and probabilities.

Page 276

Example 7 – Normal Distribution Probability

Let x have a normal distribution with = 10 and = 2. Find the probability that an x value selected at random from this distribution is between 11 and 14. In symbols, find P(11 x 14).

Page 277

Solution – Normal Distribution Probability

μ=10, σ=2

P(11 x 14) = P(0.50 ≤ z ≤ 2.00)

= Normalcdf(.5,2) = .2857874702 ≈ .2858

Using the Calculator (without converting to z scores)

Normalcdf (lower bound, upper bound, μ , σ)

-E99 and E99 are used for left tail and right tail bounds

Inverse Normal Distribution

The inverse normal probability distribution is used when we need to find z or x values that correspond to a given area under the curve. When using Table 5:

Locate the area in the body of the table If an exact area is not in the table, use the nearest

area rather than using between values. The area you use will depend on which case you

have

Page 279

Left Tail Case: Use the shaded area, A

Right Tail Case:Use 1 – A (non shaded area)

Center Case: Use the left tail

Different Cases of Inverse Normal Distributions

Page 279

A

1 – A

Using the Calculator

To find x: Hit 2nd VARS, choose 3:invNorm Enter area, μ, σ) ENTER

To find z: Hit 2nd VARS, choose 3:invNorm Enter area) ENTER

Note: The “area” you use depends on which case you have!

Example 8 – Find x, Given Probability

Magic Video Games, Inc., sells an expensive video games package. Because the package is so expensive, the company wants to advertise an impressive guarantee for the life expectancy of its computer control system. The guarantee policy will refund the full purchase price if the computer fails during the guarantee period. The research department has done tests that show that the mean life for the computer is 30 months, with standard deviation of 4 months. The computer life is normally distributed. How long can the guarantee period be if management does not want to refund the purchase price on more than 7% of the Magic Video packages?

Page 279

μ = 30months, σ = 4 months, area = 7% =

Solution – Find x, Given Probability

7% of the Computers Have a Lifetime Less Than the Guarantee PeriodFigure 6-26

invNorm(.0700,30,4) = 24.09683589

≈ 24.09 months

Interpretation The company can guarantee the Magic Video Games package for x = 24 months. For this guarantee period, they expect to refund the purchase price of no more than 7% of the video games packages.

Example 9 – Find z

Find the z value such that 90% of the area under the standard normal curve lies between –z and z.

Page 281

.90.05

invNorm(.05) = -1.644853626

z = ±1.65

How to Determine Whether Data Have a Normal Distribution

If you are not told in some why that a data set is normal or approximately normal, then you need to determine this

The following guidelines represent useful devices for determining whether or not data follow a normal distribution.1. Histogram: should be roughly bell-

shaped2. Outliers: there should not be more than 1

Page 283

Example 10 – Assessing Normality Consider the following data, which are

rounded to the nearest integer.

Page 283

a. Look at the histogram and box-and-whisker plot generated by Minitab in Figure 6-30 and comment about normality of the data from these indicators.

Example 10 – Assessing Normality

Solution:Note that the histogram is approximately normal. The box-and whisker plot shows just one outlier. Both of these graphs indicate normality.

Histogram and Box-and-Whisker PlotFigure 6-30

Assignment

Page 286 #1 – 3, 4a, 5, 11, 13,15, 19, 23, 27, 29