```7.2 Sampling Distribution of the Sampling Mean
For a finite population, the sampling distribution of the sample mean is the ________________ from all possible samples of
a _____________________.
Shape, Center, Spread of the Sampling Distribution
x
If a random sample of size n is selected from the population with a mean ___ and a population __, then:

the mean ____, of the sampling distribution of x equals the mean of the population ___:
Formula: _____________________

The standard deviation, _____, of the sampling distribution x , sometimes called the standard error of the mean,
equals the standard deviation of the population, ____, divided by the square root of the sample size n.
Formula: _____________________

The shape of the sampling distribution will be _________________ if the population is approximately normal; for
other populations, the sampling distribution becomes more ____________ as n increases. This is called the
_____________________ Theorem.
All three properties depend on the fact that the sample was selected _________________________.
Note: Rules 1 and 2 come from the rules for combining independent random variables:
x 
x 
 x  x ... x 
n
  x  x ... x 
n


 x   x  ...   x
n

 x2   x2  ...   x2
n
n  x
 x
n

n   x2
n

n   x2
 x2  x


n2
n
n
Note: Rule 3 describes SHAPE: the distribution of x is normal or approximately normal when the population is normal or
the sample is large.
Note: Rule 3 says that n needs to be “sufficiently large.” But how large is large? It depends on the original population. If it
is close to normal, n need not be large, but if it is very non-normal, n needs to be bigger. In general, the CLT can be applied
if n > 20.
1
Using the Properties of the Sampling Distribution of the Mean
1. When can I use the property that the mean of a sampling distribution of the mean is equal to the mean of the population,
x   ?
___________________________________________________.
2. When can I use the property that the standard error of the sampling distribution of the mean,
x 

n
?
You can use this formula with a population of any shape and with any sample size as long as you randomly sample
_______________________ OR you randomly sample without replacement and the sample size is less than _____ of the
________________.
3. I can only compute a probability using a z-score with I have an approximately normal distribution. When can I assume
the sampling distribution is approximately normal?
With __________________________ sample, the sampling distribution is approximately normal. If you are told the
population is approximately normal, then the sampling distribution of the mean is ______________________, no matter
what the sample size. If you are told the sample size if very large, it’s safe to assume that the sampling distribution of the
mean is __________________________________. If you had a sample size of _____ or more, you could assume a
distribution that is approximately normal.
4. Isn’t the size of the population really important? A random sample of 200 households would provide more information
about a city of size 20,000 than a city of 2,000,000.
Not really. As long as the sample is ________________________ and as long as the population is ___________________
than the sample, it doesn’t matter how large the population is.
2
Finding Probabilities using the Sample Mean
Suppose that the weights of apples are approximately normally distributed with a mean of 200 grams and a standard
deviation of 35 grams.
a. What is the probability that if you choose one apple at random, its weight is at least 220 grams?
Let x = weight ~ N(200, 35)
P(x ≥ 220) = normalcdf(220,99999,200,35) = .2839
b. What is the probability that if you choose 25 apples at random, the average weight is at least 220 g?
Since x ~ N, x ~ N(200,
35
)
25
P( x ≥ 220) = normalcdf(220,9999,200,7) = .0021 (why is this smaller than part a?)
c. What would you expect the range of weight to be for the apples chosen in a random sample of 25?
Finding Probabilities involving Sample Totals
Find the equivalent average, by dividing the total number by the sample size.
Do as above.
3
4
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