7.2. The z-Score and the Central Limit Theorem http://www.ck12.org
Using the graphing calculator the area forz=− 0 .8 is 0.2881 and forz= 1 .2 is 0. 3849
Area is: 0. 2881 + 0. 3849 = 0. 6730
(. 6730 )( 6000 ) = 4038
This means that 67.3% of the 6000 batteries lasted between 360 and 460 days.Note:Forz=− 0 .8, the area is
to the left of the mean. However, the curve is symmetrical about the mean and the value of the area forz= 0. 8
is used and added to the area ofz= 1 .2. 4038 batteries will last between 360 and 460 days. (b)z=x−μ
σ
z=320 − 400
50
z=− 1. 6
The area forz= 1 .6 is 0.4452.
0. 4452 + 0. 5000 = 0. 9452
(. 9452 )( 6000 ) = 5671
This means that 94.52% of the 6000 batteries lasted more than 320 days.Note:Forz=− 1 .6, the total area
to the right of the mean is needed. Since the total area under the curve is one, the total area on either side of
the mean is 0.5000. This area must be added to the area 0.44525671 batteries will last more than 320 days (c)z=
x−μ
σ
z=280 − 400
50
z=− 2. 4
The area forz= 2 .4 is 0.4918.
0. 5000 − 0. 4918 = 0. 0082
(. 0082 )( 6000 ) = 49
This means that 0.82% of the 6000 batteries lasted less than 280 days.Note:Since the total area to the left of
z=− 2 .4 is required, the area forz= 2 .4 is subtracted from 0.500049 batteries will last less than 280 days
1.μ=25 The population mean of 25 was given in the question.- ̄x= 25 .5 The sample mean is 25.5 and is determined by using 1−Vars Staton the TI-83.
3.σ=4 The population standard deviation of 4 was given in the question.
4.Sx= 3 .47 The sample standard deviation is 3.47 and is determined by using 1−Vars Staton the TI-83.
5.μx ̄=25 A property of the Central Limit Theorem.
σx ̄=
σ
√
n
σx ̄=
4
√
40
σx ̄= 0 .63 Central Limit TheoremVocabulary
Central Limit Theorem
An important result in statistics, stating that the shape of the sampling distribution of the sample mean becomes
more normal asnincreases.