Transcript Probability Distribution - Department of Statistics and

Math 1107
Introduction to Statistics
Lecture 11
The Central Limit Theorem
MATH 1107 – The Central Limit Theorem
Often we cannot analyze a population
directly…we have to take a sample. What are
some of the reasons we sample?
MATH 1107 – The Central Limit Theorem
After we take a sample, in order to make
inferences onto the population, we have to
assume the data is normally distributed. What
if our population is not normal? Do we have a
problem?
http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html
MATH 1107 – The Central Limit Theorem
Important concepts to remember about the Central Limit
Theorem:
1. The distribution of sample means will, as the sample size
increases approach a normal distribution;
2. The mean of all sample means is the population mean;
3.The std of all sample means is the std of the population/the
SQRT of the sample size;
4. If the population is NOT normally distributed, sample sizes
must be greater than 30 to assume normality;
5. If the population IS normally distributed, samples can be of
any size to assume normality (although greater than 30 is always
preferred).
MATH 1107 – The Central Limit Theorem
Example of Application (Page 262):
If a Gondola can only carry 12 people or 2004 lbs
safely, there is an inherent assumption that each
individual will weigh 167 lbs or less. Men weigh on
average 172 lbs, with a std of 29 lbs.
Assume that any selection of 12 people is a sample
taken from an infinite population. What is the
probability that 12 randomly selected men will have a
mean that is greater than 167 lbs?
MATH 1107 – The Central Limit Theorem
Because we assume that weight is normally distributed (it
almost always is), we can comfortably use a sample less than
30. We can also assume that the mean of our samples will be
the same as our population mean, and that the std of our
sample is the same as the population std/SQRT of the sample
size 29/SQRT(12) or 8.372
Now, we can calculate a Z-score…
Z = 167-172/8.372. This equals -.60. Or 73%.
What is the interpretation of this figure?
MATH 1107 – The Central Limit Theorem
Example of Application (Page 265):
Assume that the population of human body
temperatures has a mean of 98.6F. And, the std is
.62F. If a sample size of n=106 is selected, find the
probability of getting a mean of 98.2F or lower.
Here, we don’t know how the population is
distributed, but because the sample size is greater
than 30, it does not matter…we can assume the
distribution of the sample is normal.
MATH 1107 – The Central Limit Theorem
Again, we assume that the sample means will be the
same as the population mean (98.6) and the std of
the samples is the same as the std of the
population/SQRT of the sample size (.62/SQRT(106)).
Now, we can calculate a Z-score:
Z=98.2-98.6/.06022 = -6.64
What does this mean?