Topics for Today. Sampling Distribution. The Central Limit Theorem. Stat203 Page 1 of 28 Fall 2011 Week 5 Lecture 2

Transcription

1 Topics for Today Sampling Distribution The Central Limit Theorem Stat203 Page 1 of 28

2 Law of Large Numbers Draw at from any population with mean μ. As the number of individuals, the mean,, of the sample gets to the mean μ of the population. Stat203 Page 2 of 28

3 LLN is the foundations of businesses such as casinos and insurance companies. Winnings or losses of a gambler on a few plays are uncertain, which is why gambling is exciting. It is only in the long run that the mean is predictable. The house plays tens of thousands of times so, unlike the individual gambler, they can count on the long run regularity described by the law of large numbers. Stat203 Page 3 of 28

4 Large means large. Fallacy of the LLN Gamblers often succumb to believing the LLN will help them to predict the next occurrence of an event (eg: red, blackjack, etc). This is not the case. Consecutive runs are independent. Google this for more: Monte Carlo 1913 Another poor example: A mathematician always takes a bomb on board an airplane reasoning that the odds of a bomb on a plane are small so the odds of two bombs on a plane are virtually zero. Stat203 Page 4 of 28

5 Sampling Distributions The law of large numbers us that if we measure enough, the will eventually get very close to μ. What more can we say about the of? Suppose we took a of 10 individuals from a population and calculate. What does the of look like? Stat203 Page 5 of 28

6 Question: What would happen if we took of 10 individuals from a population? Take a large number of samples of size 10 from the population Calculate for each sample Make a histogram of these values of examine it. and Stat203 Page 6 of 28

7 Example: Constructing a Sampling Distribution Extensive studies have found that the odor threshold of adults follows roughly a distribution with μ = 25 micrograms/litre and a σ = 7 micrograms/litre. With this information, we can simulate many runs of our study with individuals drawn at from the population. The next figure illustrates the process: the authors took of, found the mean odor threshold and made a histogram of these 1000 ʼs. Stat203 Page 7 of 28

8 Stat203 Page 8 of 28

9 What can we say about the,, and of this distribution? The histogram shows how would behave if we drew many samples; the of the statistic. The figure on the next page compares the mean odor threshold to the distribution of odor thresholds for a single adult. Stat203 Page 9 of 28

10 Stat203 Page 10 of 28

11 The is the that would emerge if we look at samples of 10 individuals from our population. So, connecting to some earlier ideas: The Probability Distribution is the underlying theoretical distribution of for an entire population. The Sampling Distribution is the underlying theoretical distribution of the (like the mean) for all possible samples from a population. Stat203 Page 11 of 28

12 Mean and Standard Deviation of Suppose that is the mean of a of size n drawn from a large population with mean μ and standard deviation σ. Then the of the of standard deviation is. is μ and its this is true of the underlying! Stat203 Page 12 of 28

13 Mean of the Sampling Distribution The of ʼs sampling distribution is always the same as the mean, μ, of the population. The sampling distribution of at μ. is In repeated sampling, will sometimes fall above the true value of the population parameter μ and sometimes below, but there is tendency to or. We say is an estimator of μ. Stat203 Page 13 of 28

14 Standard Deviation of the Sampling Distribution How close is to μ? Averages are than individual observations. The standard deviation of is than the standard deviation of the individuals. Standard deviation of =. The results of are variable than the results of small samples. Stat203 Page 14 of 28

15 If, the standard deviation of will be, and almost all samples will give values of that are to μ. Note that to cut the standard deviation of in half we must take four times as many observations, not just twice as many. Stat203 Page 15 of 28

16 Normal Distributions If a variable measured on a population has a normal distribution, then the distribution of the sample means generated by a random sample has a normal distribution. If X is with mean μ and standard deviation σ, then the distribution of the sample mean of a of n observations has a with mean μ and standard deviation. Stat203 Page 16 of 28

17 What happens when the population is? As the sample size, the distribution of shape, it looks less like that of the population and more like a distribution. This is true no matter what shape the population distribution has. This famous fact is called: The Central Limit Theorem Stat203 Page 17 of 28

18 The Central Limit Theorem Draw a of size n from any population with finite mean μ and standard deviation σ. When n is, the of the sample mean is. is approximately normal with mean μ and standard deviation. How large a sample? More are required if the shape of the population distribution is from normal. Stat203 Page 18 of 28

19 Stat203 Page 19 of 28

20 The Central Limit Theorem in Action The above figure shows how the central limit theorem works for a fairly non-normal population. The first figure a) displays the probability distribution of a single individual, that is, of the entire population. The distribution is skewed with the most probable outcomes near 0. The mean μ of this distribution is 1, and its standard deviation σ is also 1. This distribution is called an distribution. Stat203 Page 20 of 28

21 Exponential distributions are used as models for the lifetime in service of electronic components as well as the time required to serve a customer or repair a machine. Stat203 Page 21 of 28

22 The next three figures are the density curves of the sample means of size _,, and observations from this population. As the sample size n, the shape of the distributions becomes. The mean μ = 1, and the standard deviation decreases taking the values. The density curve for 10 observations is positively skewed but is close to resembling the normal distribution with mean μ = 1 and standard deviation σ = =.32. The density curve for a sample of 25 observations is even closer to the distribution. Stat203 Page 22 of 28

23 We can clearly see the contrast in shapes between the population distribution and the distributions of the means. Try some others: Stat203 Page 23 of 28

24 Example: Maintaining air conditioners The time (X) that a worker requires to perform preventative maintenance on an air conditioning unit is governed by the exponential distribution. The mean repair time is μ =1 hour and the standard deviation σ = 1 hour. If your company operates 70 of these units, what is the probability that their average maintenance time exceeds 50 minutes? Stat203 Page 24 of 28

25 Example: Flaws in carpets The number of defects per square meter in a type of carpet material varies with mean μ = 1.6 flaws/m 2 and standard deviation σ = 1.2 flaws/ m 2. The population distribution is not normal since the number of flaws is a count. An inspector looks at 200 square meters of the material and records the number of flaws per square meter and calculates the sample mean. Use the central limit theorem to find the probability that the mean number of flaws is greater than 2 per square meter. Stat203 Page 25 of 28

26 Stat203 Page 26 of 28

27 Todayʼs Topics Sampling Distribution of the mean - has the same mean as the population - has standard deviation is the population standard deviation divided by the squareroot of n Central Limit Theorem - Says that no matter what the underlying probability distribution, the sampling distribution of the mean will be Normal Stat203 Page 27 of 28

28 Reading for next lecture Chapter 6 Confidence Intervals Stat203 Page 28 of 28