6.2 Standard Units and Areas under the Standard Normal Distribution

6.3 Area Under Any Normal CurveChapter 6: Normal Curves and Sampling DistributionsNormal Distribution AreasIn many applied situations, the original normal curve is not the standard normal curve. How to Work with Normal DistributionsTo find areas and probabilities for a random variable x that follows a normal distribution with a mean and standard deviation , convert x values to z values using the formula

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

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Example 7 Normal Distribution ProbabilityLet 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 277Solution Normal Distribution Probability

=10, =2

P(11 x 14) = P(0.50 z 2.00)

= Normalcdf(.5,2) = .2857874702 .2858Using the Calculator (without converting to z scores)Normalcdf (lower bound, upper bound, , )

-E99 and E99 are used for left tail and right tail boundsInverse Normal DistributionThe 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 tableIf 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 havePage 279Left Tail Case: Use the shaded area, ARight Tail Case:Use 1 A (non shaded area)Center Case: Use the left tailDifferent Cases of Inverse Normal DistributionsPage 279

A1 A

Using the CalculatorTo find x:Hit 2nd VARS, choose 3:invNormEnter area, , ) ENTERTo find z:Hit 2nd VARS, choose 3:invNormEnter area) ENTER

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

Example 8 Find x, Given ProbabilityMagic 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 = 30months, = 4 months, area = 7% = Solution Find x, Given Probability7% of the Computers Have a Lifetime Less Than the Guarantee PeriodFigure 6-26

invNorm(.0700,30,4) = 24.09683589 24.09 monthsInterpretation 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

.90.05invNorm(.05) = -1.644853626z = 1.65How 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 thisThe following guidelines represent useful devices for determining whether or not data follow a normal distribution.Histogram: should be roughly bell-shapedOutliers: there should not be more than 1Page 28312Example 10 Assessing NormalityConsider the following data, which are rounded to the nearest integer.

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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 NormalitySolution: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

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