Download The Standard Normal Distribution (continued)

Math 150
Lecture Notes
10/14
The Standard Normal Distribution (continued)
Ex: Find the following probabilities
(a) P (Z > −1.21)
(b) P (−1 < Z < 1.35)
Ex: Find the value of a such that
(a) P (Z > a) = 0.05
(b) P (Z > a) = 0.75
(c) P (−a < Z < a) = 0.70
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.
Ex: The amount of sugar in energy drinks is approximately normally
distributed with a mean of 40 grams and a standard deviation of 7
grams.
(a) What is the probability that a randomly selected energy drink
will contain more than 56 grams of sugar?
(b) What is the probability that a randomly selected energy drink
will contain between 30 grams and 50 grams of sugar?
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Ex: The speed of cars in the 91 Freeway is normally distributed with
a mean of 65 mph. If 3% of cars travel faster than 85mph,what is the
standard deviation of this distribution?
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The Central Limit Theorem (CLT)
Regardless of the distribution as long as the sample size is at least
30, x is normally distributed with mean µ and standard deviation
√
σ/ n
Ex: The amount of caffeine in ∅rganic coffee has a mean of 60 mg
and a standard deviation of 8 mg. If you pick a random sample of 36
cups of coffe,
(a) What is the probability that the sample mean will contain less
than 57 mg of caffeine?
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Z0.05 = 1.64,
Z0.01 = 2.33,
Confidence Intervals for
Z0.025 = 1.96 , Z0.005 = 2.58
µ(σ
known )
Procedure
Step 1: Find
Step 2: Find
Step 3: Find
Step 4: Write
α/2, Zα/2
σ
E = Zα/2 √
n
X ±E
X −E <µ<X +E
Ex: A study of 36 taxi drivers shows that the average time they wait
for a customer is 65 minutes with a standard deviation of 10 minutes.
a) Find a 95% confidence interval for the true waiting time of all
taxi drivers.
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b) Find a 99% confidence interval for the true waiting time of all
taxi drivers.
c) Which interval is smaller? Explain.
Ex: Dr. ϑ wants to find out the average number of hours that students spend doing Mathematics homework per week. He conducts of
survey of 100 students. The average is 4.2 hours and the standard
deviation is 2.2 hours.
a) Find a 99% Confidence interval for the real number of hours that
students study per week?
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b) Juan claims that he studies on average 6 hours per week. Is that
reasonable?
Sample Size:
n=
!
Zα/2σ
E
"2
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Confidence Intervals for
p
Procedure
Step 1: Find
α/2, Zα/2
!
p̂(1 − p̂)
n
Step 2: Find
E = Zα/2
Step 3: Find
p̂ ± E
p̂ − E < p < p̂ + E
Step 4: Write
x
p̂ =
n
Ex: Nacho is interested in finding out the percentage of Angelinos
that believe in the existence of Santa Claus. He conducts of survey
with 200 people, of which 140 said that they believe in the existence
of Santa Claus.
a) Find a 95% confidence interval for the true proportion of Angelinos
that believe in the existence of Santa Claus.
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b) How large of a sample is needed in order to have a 2% margin of
error?
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