Lecture 11 The Central Limit Theorem Math 1107 Introduction to Statistics.

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Presentation transcript:

Lecture 11 The Central Limit Theorem Math 1107 Introduction to Statistics

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?

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 Now, we can calculate a Z-score… Z = / This equals 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= / = What does this mean?