Transcript Section 6.2 Third Day Combining Normal RVs

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Chapter 6: Random Variables
Section 6.2
Transforming and Combining Random Variables
The Practice of Statistics, 4th edition – For AP*
STARNES, YATES, MOORE
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Combining Normal Random
Variables
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One of the skills you need to learn from this section is
combining two independent normal random variables and
finding probabilities.
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Find the combined mean and standard deviation, and then
work the problem as you would any normal curve probability
(find the Z-score).
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Example
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Suppose women’s heights are normally distributed with a mean
of 64” and a standard deviation of 2.5”.
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Suppose men’s heights are normally distributed with a mean of
69” and a standard deviation of 2.25”.
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What is the probability that the difference in height between a
randomly chosen male and a randomly chosen female is more
than 12 inches?
Normal Random Variables
Mr. Starnes likes between 8.5 and 9 grams of sugar in his hot tea.
Suppose the amount of sugar in a randomly selected packet follows a
Normal distribution with mean 2.17 g and standard deviation 0.08 g. If
Mr. Starnes selects 4 packets at random, what is the probability his tea
will taste right?
Let X = the amount of sugar in a randomly selected packet.
Then, T = X1 + X2 + X3 + X4. We want to find P(8.5 ≤ T ≤ 9).
8.5  8.68
9  8.68
 1.13
and+2.17
z = 8.68  2.00
µT = µX1 + µX2 + µX3 + µzX4 = 2.17 + 2.17
+ 2.17
0.16
0.16
2
2
2
2
2
T2  X2  X2  X2  P(-1.13
(0.08)
 0.0256
≤ Z≤(0.08)
2.00) =(0.08)
0.9772 –
0.1292
= 0.8480
X  (0.08)
There is about an 85% chance Mr. Starnes’s
T  0.0256 
0.16
tea will taste right.
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Transforming and Combining Random Variables
Example
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 Combining