Transcript Sampling_Distributions

Sampling Distributions
Adapted from Exploring Statistics with the TI-83 by Gail
Burrill, Patrick Hopfensperger, Mike Koehler
A company wants to know what percent of consumers like the
ads for their product.
A TV network would like information on the proportion of the
TV viewing public that watches their programming.
In both cases, information is needed from a large number
people, but it would not be possible to ask the entire
populations for their opinions.
Samples may be made and the information from the samples
may be used to make inferences about the populations.
Let’s simulate a sample of consumers polled about whether they
liked an advertising campaign to promote a new shampoo.
Assume that 40% liked the ads and 60% disliked the ads.
We will use the TI-83 command randBin to simulate a
sample of 20 persons, 40% of whom liked the ads.
We will record the number who liked the ads.
(RandBin is found under the <MATH> <PRB> menu.)
9 out of 20 persons liked the ads.
Now we can try this on a larger scale.
Run the simulation 100 times!
An easy way to do this is to specify 100 runs and save the
results to L1.
Make a histogram of the results and
calculate the mean and standard
deviation.
The mean is 8.09 and
the sample standard
deviation is 2.374.
A boxplot can also help to summarize the sampling distribution.
Here is a “90% boxplot” of the results of the 100 trials,
followed by the distribution of results. 2 I
3
4
5
6
7
8
9
10
11
12
13
14
III
IIIII IIIII I
IIIII IIIII II
IIIII IIIII IIIII II
IIIII IIIII IIII
IIIII IIIII IIII
IIIII IIII
IIIII IIII
IIIII IIII
I
In a 90% boxplot, at least 90% of the 100 trials must fall on or
within the box. Approximately 5%, or fewer of the 100 trials
should lie on each segment outside the box (in the whiskers).
In this case, 95% are within the box, as the high and low values
of 12 and 5, occurred 9 and 11 times, respectively.
The TI-83 does not have a built-in feature for making a 90%
boxplot so I have provided a program, called BOXPLOTS.
You may either enter it yourself or link your calculator and
download it directly from the Media Center..
At least 90% of the sample trials lie
between 5 and 12.
The definition of a “likely event” is one that is likely to
happen in at least 90% of the trials.
90% is chosen arbitrarily here, but is often used in statistics.
In our case, we might say that in a
sample of 20 consumers asked about
the new shampoo ads, it is likely that
between 5 and 12 will indicate that
they like the ads.
It is unlikely, for instance, that 15 out of 20 randomly selected
consumers will like the ads.
Problem: 45% of people in the US have type O+ blood.
Simulate 100 random samples of 40 people. Find a 90% boxplot
of the sample trials. Based on this boxplot, what outcomes
would be likely for any random sample of 40 people?
In 94 out of 100 simulations between 12 and 23 people had
O+ blood. (The statistic 94 was found by examining the
data in L1, not shown.)
Based on the distribution, in at least 90%of the random
samples of 40 people, it would be likely to have anywhere
from 12 to 23 people with type O+ blood.
Now you will have a chance to try a few problems in class.
THE END