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Happiness comes not from material wealth but less desire.
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Means and Proportions as
Random Variables
Sampling
distribution
Normal curve
approximation
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Definitions



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A statistic is a numerical summary of a
sample. Its value may differ for different
samples.
A parameter is a numerical summary of a
population, which is a (unknown) constant.
The sampling distribution of a statistic is
the distribution of possible values of the
statistic for repeated random samples of the
same size taken from a population.
Sample Proportion


p̂
In a random sample of size n, the sample
proportion p̂ is the proportion of, say women,
out of the sample. For example, if there are 8
women in a sample of 10 students, then the
sample proportion for women is p̂ =8/10=0.8.
The population proportion is denoted as p.
Q: Identify each of the following as a statistic or
a parameter: 1) p
2) p̂ .
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Sampling Distribution of

Mean:

Standard deviation:
E ( pˆ )  p
StD( pˆ ) 

p(1  p)
n
Standard error, the estimated standard deviation:
StE( pˆ ) 
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p̂
pˆ (1  pˆ )
n
Normal Curve Approximation for
Sample Proportion

The sampling distribution of p̂ can be
approximated by a normal distribution with
the mean p and standard deviation p(1  p)
n
WHEN
both np > 5 and n(1-p) > 5.

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Empirical Rule
Example: Lottery

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Suppose that probability is p=0.2 that a
person purchasing an instant lottery
ticket wins money, and this probability
holds for every ticket purchased.
Consider all random samples of 64
purchased tickets, and let p̂ be the
sample proportion of winning tickets in a
sample of 64 tickets.
• What is the average (sample) proportion of winning
tickets in 64 randomly selected tickets?
• What is the standard deviation of the sample proportion p̂ ?
• What is the probability that there are more than 32
winning tickets in 64 randomly selected tickets?
• Using Empirical Rule to fill the blanks:
In about 95% of all random samples of 64
purchased tickets, the proportion of winning tickets
will be between ____ and ____.
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Sample Mean X
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
The sample mean of a random sample of size
n is the average in that sample.

Let m and s denote the population mean and
standard deviation of the population sampled.
Sampling Distribution of X

Mean:

Standard deviation:
E (X )  m
SD ( X )  s / n

Standard error, the estimated standard deviation:
SE ( X )  s / n
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Normal Curve Approximation for
Sample Mean
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
The sampling distribution of the sample mean,x
from a random sample of size n (where n is at
least 30) can be approximated by a normal
distribution with the mean m and standard
deviation s n.

Empirical Rule
Example: Speed at 880
Vehicle speeds at highway 880 are believed to have
mean m60 mph. The speeds for a randomly selected
sample of n=36 vehicles at highway 880 were
recorded and its standard deviation is s=6 mph.
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
What is the mean and standard error of (the sampling
distribution of) sample mean?

Use the Empirical Rule to fill the blanks: For a
random sample of 36 vehicles, there is about a 68%
chance that the mean vehicle speed in the sample
will be between ____ and ____.