Download Section 7.2 sampling distribution of sampling mean discussion

Section 7.2 Sampling Distribution of the Sample Mean DISCUSSION
When sampling distributions are created, often a _____________ shape is
formed.
The upper left hand figure gives data for how many cigarettes per day are
smoked by 4893 smokers. The other three graphs are sampling distributions
from this data set
where the mean of
random sets of 5, 20,
and 50 are plotted.
How does the mean
compare with the
population mean?
_______________
How does the shape
change from the
original data set as
the sample size, n,
increases?
_________________
How does the
standard deviation
change as the sample
size, n, increases?
_______________________
In summary, three patterns that you should notice:
1.) As the sample size, n, increases, the shape of the sampling distribution
becomes _____________, even if the original population is ____________.
2.) The means of all sampling distributions are the same as the ___________
for the ___________________.
3.) The _______________________ of the sampling distributions __________
as _____ increases.
It is sometimes confusing to keep track of whether we are talking about means
and standard deviations of the ______________, a _______________, or the
____________________________. The table below gives the symbolism
commonly used:
Population Sample Sampling
Parameter Statistic Distribution
Mean
Standard
Deviation
Size
If a random sample of size n is selected from a population with a mean _____
and standard deviation _______,
Properties of the Sampling Distribution of the Sample Mean
Feature
Description
Symbol
Mean, ____, of sampling distribution equals
Center
the mean of the population, _____.
Standard deviation (_________________________),
Spread
_____, of the sampling distribution of _____ equals the
standard deviation of the population, _____, divided by the
square root of the sample size, _____.
The shape of the sampling distribution is approximately
Shape
____________ if the population is approximately normal.
For other populations, the distribution becomes more
______________ as ______ _____________.
The property where by as _____ increases the shape becomes more
____________ is called the _______________________________.
The standard error formula, _____________, is important because we can now
compute the standard error without having to do a _____________________.
We can solve problems involving the _______________________ by using the
properties we have just observed along with ______________________.
Example: We will look at the proportion of households owning different
numbers of cars, the data from section 6.1.
Vehicles per Proportion of
Household Households
0
0.088
1
0.332
2
0.385
3
0.137
4
0.058
The mean of the population, _____, is:
0(0.088) + 1(0.332) + 2(0.385) + 3(0.137) + 4(0.058) = ________
The standard deviation, _______, of the population is:
0  1.7452 0.088  1  1.7452 0.332  2  1.7452 0.385  3  1.7452 0.137  4  1.7452 0.058
= __________
What is the probability that in a random sample of 25 households, the average
number of cars will be less than 2?
What average numbers of cars of reasonably likely in a random sample of 25
households?
Finding Probabilities Involving Sample Totals
So far we have looked at probabilities of having in a sample of a certain size an
average of some value or less. We might also want probabilities of having a
total number in a sample rather than an average value. For example, instead of
having the probability of a sample of 25 households having an average of 2
cars or less, we might look at the probability of a sample of 25 households
having a total of 50 cars or less.
There are two possible ways to handle this situation:
First
Find the equivalent average number of cars, ______, by ________________ the
total number of cars, _______, by the sample size, ________:
Then use the same procedure as above.
Second
Change the formulas from average numbers to total numbers. The formula
changes are:
Properties of the Sampling Distribution of the Sum of a Sample
 The mean of the sampling distribution of the ________ is:
 The ___________________ of the sampling distribution of the sum
is:
 The shape of the sampling distribution is normal if the population is
normal and becomes more normal if the population shape is not
normal.
Example: What is the probability of a sample of 25 households having a total
of 50 cars or less?
We can also look at reasonably likely totals:
In a random sample of 25 households, what total numbers of cars are reasonably
likely?