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Stat 104, Section 5 Handout Solutions
Benny, Thursday, 4:00pm, SC 309
Key concepts
•
Continuous Random Variables
•
Le Normal Distribution
•
Sample distribution of
¯
X
(implicit dependence on the central limit theorem)
•
tdistirbution
•
Conﬁdence intervals
Practice Problems
1.
The daily high temperature for the ﬁrst week in October in Boston is approximately Normally
distributed with an average high of 66 F and a standard deviation of 7 F.
a.
What is the probability that a day in the ﬁrst week of October would have a high of 57.8 F
or colder?
Let
X
be a variable to represent the temperature for a day inthe ﬁrst week of October. We
want
P
(
X
≤
57
.
8). Assuming normality, we calculate the zscore
z
=
X

¯
X
σ
X
=
57
.
8

66
7
=

1
.
17
Referring to the zvalue probability table, we get
P
(
z <

1
.
17)
≈
12%
b.
What high temperature needs to be achieved in order to be in the bottom 1% of the daily
high temperatures in the ﬁrst week of October?
We do the reverse of part a. Speciﬁcally, we ﬁnd
z
low
satisfying
P
(
z < z
low
) =
.
01
And ﬁnd
z
low
=

2
.
33. Converting this into the temperature scale with the standard deviation,
we have
T
low
= 66

2
.
33
*
7 = 49
.
7
1
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This note was uploaded on 03/27/2012 for the course STATS 104 taught by Professor Michaelparzen during the Fall '11 term at Harvard.
 Fall '11
 MichaelParzen
 Central Limit Theorem, Normal Distribution

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