Algebra 1 Common Core Student Edition, Grade 8-9

.

Co n cep t Byt e

Use W ith Lesson 12-8

Normal

Dist ribut ions

S - ID .A .4 Use the mean and standard deviation o f a

data set to fit it to a normal distribution and to estimate

population percentages...

MP 5

Common Core State Standards

A normal distribution shows d ata th at vary ran d o m ly from th e m e a n in a p a tte rn of

a symmetric, bell-shaped curve called a normal curve. A n o rm al distribution has the

following properties.

MATHEMATICAL

PRACTICES

• About 68 % of the d ata fall w ithin one stan d ard deviation of
  th e m ean.


• About 95% of th e d ata fall w ithin two stan d ard deviations of
  the m ean.


• About 99.7% of th e d ata fall w ithin three stan d ard deviations
  of th e m ean.

-3

deviations

1 mean

deviation

I

deviations

+ 1 +3

deviation deviations

+2

Ex a m p l e 1

The weights o f pum pkins a farmer harvests on e year are norm ally distributed with a

m ean of 40 lb and a standard deviation o f 5 lb.

0 Find the percent o f the pum pkins that w eigh betw een 35 lb and 50 lb.

Draw a norm al curve for th e distribution. Note th a t 35 lb is 1 stan d ard deviation

below th e m e a n a n d 50 lb is 2 stan d ard deviations above th e m ean.

-3

deviations

-1 mean -1

deviation

: r 2 i

i d e v i a t i o n s

+3

deviation deviations

+2

25 30 35 40 45 50 55

Weight (lb)

Add th e percents b etw een 35 lb an d 50 lb on th e n orm al curve.

34% + 34% + 13.5% = 81.5%

About 81.5% of th e p u m pkins w eigh betw een 35 lb an d 50 lb.

Find the percent o f the pum pkins that weigh betw een 33 lb and 43 lb.

N either 33 lb n o r 43 lb is a w hole n u m b e r of stan d ard deviations from th e m ean,

so you ca n n o t use th e n o rm al curve draw n in p a rt (A). Instead, use a graphing

calculator th a t com putes p ercents associated w ith n o rm al distributions. One

m odel of calculator uses th e following com m and.

normalcdf(/ojeer bound, upper bound, mean, standard deviation)

For this problem, enter normalcdf(33,43,40, 5). I h e calculator retu rn s a p ercen t

expressed as a decim al: ab o u t 0.645. This m ean s th a t a b o u t 64.5% of the pum pkins

weigh betw een 33 lb an d 43 lb.