Transcript Review Question!!!

10.1: Confidence Intervals
– The Basics
Review Question!!!
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If the mean and the standard deviation of a
continuous random variable that is normally
distributed are 26 and 6 respectively, find an
interval that contains 95% of the distribution.
A) ( 14, 38)
B) ( 8, 44)
C) ( 20, 32)
D) ( 6, 46)
Review Question!!!
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A wholesale distributor has found that the amount of a
customer’s order is a normal random variable with a
mean of $200 and a standard deviation of $50. The
distributer takes a sample of 25 orders, what is the mean
of the sampling distribution?
A) 50
B) 200
C) 250
D)10
Review Question!!!
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A wholesale distributor has found that the amount of a
customer’s order is a normal random variable with a
mean of $200 and a standard deviation of $50. The
distributer takes a sample of 25 orders, what is the
standard deviation of the sampling distribution?
A) 50
B) 200
C) 250
D)10
Review Question!!!
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A wholesale distributor has found that the amount of a
customer’s order is a normal random variable with a
mean of $200 and a standard deviation of $50. What is
the probability that a sample of 25 orders is within
$20 of the mean?
A) 0.997
B) 0.31
C) 0.954
D) 0.5
Introduction
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Is caffeine dependence real?
What proportion of college students engage in
binge drinking?
How do we answer these questions?
Statistical inference provides methods for
drawing conclusions about a population from
sample data.
Ex 1: IQ and Admissions
Harvard’s admissions director proposes using
the IQ scores of current students as a marketing
tool. The director gives the IQ test to an SRS of
50 of Harvard’s 5000 freshmen. The mean IQ
score is 𝑥 = 112. What can the director say
about the mean score μ of the population of all
5000 freshmen?
Ex 1: IQ and Admissions
Ex 1: IQ and Admissions
Ex 2: Estimation in Pictures
.
Ex 3: IQ Conclusion
Ex 3: IQ Conclusion
Confidence Interval & Level
25 samples from the same population gave these 95% confidence intervals. In
the long run, 95% of all samples given an interval that contains the population
mean μ.
Conditions for Constructing a
Confidence Interval for μ
Ex 4: Finding z (Using Table A)
Most Common Confidence Levels
Confidence
Level
Tail Area
Z*
90%
0.05
1.645
95%
0.025
1.960
99%
0.005
2.576
Critical Values
Confidence Interval for a Population Mean (σ Known)
When choosing an SRS from a population
(having unknown μ and known σ), the level
C confidence interval for μ is:
xz
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n
= Estimate ± Margin of Error
= Estimate ± (Critical Value of z) (Standard Error)
Ex 5: Video Screen Tension
A manufacturer of high-resolution video terminals must
control the tension on the mesh of fine wires that lies
behind the surface of the viewing screen. Careful study
has shown that when the process is operating properly, the
standard deviation of the tension readings is σ = 43 mV.
Here are the tension readings from an SRS of 20 screens
from a single day’s production:
269.5
297.0
269.6
283.3
304.8
280.4
233.5
257.4
317.5
327.4
264.7
307.7
310.0
343.3
328.1
342.6
338.8
340.1
374.6
336.1
Ex 5: Video Screen Tension
CI  x  z
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n
Step 2: Plan Cont.:
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SRS? We are told that the sample was an SRS.
Normality? The sample size is too small to use the
central limit theorem (n=20), but the boxplot of the sample
data is approximately Normal (no severe skew or outliers)
We can assume that the sampling distribution is
approximately Normal.
Independence? We must assume that at least (10)(20) =
200 video terminals were produces on this day.
xz
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n
43
306.3  1.645
20
306.3  15.8
(290.5 , 322.1)
Ex 5: Video Screen Tension
Step 4: Interpretation: Interpret your results
in the context of the problem.
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We are 90% confident that the true mean
tension in the entire batch of video terminals
produced that day is between 290.5 and 322.1
mV.
How Confidence Intervals Behave
We select the confidence interval, and
the margin of error follows…
We strive for HIGH confidence and a
SMALL margin of error.
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HIGH confidence says that our method
almost always gives correct answers.
SMALL margin of error says that we have
pinned down the parameter quite precisely.
How Confidence Intervals Behave
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Consider margin of error…

z
n
The margin of error gets smaller when…
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z gets smaller. To accept a smaller margin of error, you
must be willing to accept lower confidence.
σ gets smaller. The standard deviation σ measures the
variation in the population.
n gets larger. We must take four times as many
observations in order to cut the margin of error in half.
Sample Size for a Desired Margin
of Error
To determine the sample size that will yield a confidence
interval for a population mean with a specified margin of
error, set the expression for the margin of error to be less
than or equal to m and solve for n:
z
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n
m
Ex 2: How Many Monkeys?
Researchers would like to estimate the mean
cholesterol level μ of a particular variety of monkey
that is often used in lab experiments. They would like
their estimate to be within 1 mg/dl of the true value of
μ at a 95% confidence level. A previous study
indicated that σ = 5 mg/dl. Obtaining monkeys is timeconsuming and expensive, so researchers want to
know the minimum number of monkeys they will need
to generate a satisfactory estimate.
We must round
up!!! We need 97
monkeys to
estimate the
cholesterol levels
to our satisfaction.
z
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n
1
5
1.96
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n
(1.96)(5)  n
9 .8  n
96.04  n
Some Cautions
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Read the “Cautions” on p.636 - 637
BIG CAUTION
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We are 90% confident that the true
mean tension in the entire batch of
video terminals produced that day is
between 290.5 and 322.1 mV.
These number were calculated by a
method that give correct results in 95% of
all possible samples.
We cannot say that the probability is 90%
that the true mean falls between 290.5 and
322.1. No randomness remains after we
draw one particular sample and get from it
one particular interval.
The true mean either is or is not between
290.5 and 322.1.
The probability calculation of statistical
inference describes how often the METHOD
gives correct answers.