Chapter 7 Sampling Distributions Mc GrawHillIrwin Copyright 2009

Sample from Population Random Sample of size n Sample x Parameter mean µ inference 7 -2

Sampling Distribution of the Sample Mean The sampling distribution of the sample mean x of size n is the probability distribution of the population of the sample means obtainable from all possible samples of size n from a population. 7 -3

General Conclusions • Continued The mean of all possible sample means equals the population mean – That is, m = mx • The variance s 2 x of the sampling distribution of x is • The standard deviation sx of the sampling distribution of x is 7 -4

Sample from Population Random Sample of size n Parameter mean µ and standard deviation σ. x has mean mx=m, and standard deviation 7 -5

General Conclusions 1. If the population of individual items is normal, then the population of all sample means, x‘s, is also normal with mean mx=m and variance or equivalently 2. Suppose the population has mean m and standard deviation σ. Even if the population of individual items is not normal, but the sample size is big enough (n ≥ 30), the distribution of x is approximately normal with mx=m and (Central Limit Theorem) 7 -6

Example: Effect of Sample Size The larger the sample size, the more nearly normally distributed is the population of all possible sample means Also, as the sample size increases, the spread of the sampling distribution decreases 7 -7

Properties of the Sampling Distribution of the Sample Mean #3 • The standard deviation sx of the sampling distribution of x is • That is, the standard deviation of the sampling distribution of x is – Directly proportional to the standard deviation of the population – Inversely proportional to the square root of the sample size 7 -8

How Large? • How large is “large enough? ” • If the sample size is at least 30, then for most sampled populations, the sampling distribution of sample means is approximately normal – Here, if n is at least 30, it will be assumed that the sampling distribution of x is approximately normal • If the population is normal, then the sampling distribution of x is normal regardless of the sample size 7 -9

Example 7. 2: Car Mileage Statistical Inference • Suppose for the Chapter 3 mileage example, x = 31. 56 mpg for a sample of size n=50 and suppose the population standard deviation σ = 0. 8 • If the population mean µ is exactly 31 and standard deviation is 0. 8, what is the probability of observing a sample mean that is greater than or equal to 31. 56? 7 -10

Example 7. 2: Car Mileage Statistical Inference #2 • Calculate the probability of observing a sample mean that is greater than or equal to 31. 6 mpg if µ = 31 mpg – Want P(x > 31. 56 if µ = 31 σ=0. 8 for the population) • Since the condition for Central Limit Theorem are satisfied, we can use normal distribution to approximate the probability. To find out the normal distribution, we need to decide the mean and standard deviation: mx=m=31 7 -11

Example 7. 2: Car Mileage Statistical Inference #3 • Then • The distribution of x bar has mean 31 and standard deviation 0. 113 • But z = 4. 96 is off the standard normal table, 3. 99, P(Z<4. 96)=0. 99997 • The largest z value in the table is 3. 99, which has a right hand tail area of 7 -12

Central Limit Theorem • Now consider sampling a non-normal population • Still have: mx=m and sx=s/ n – Exactly correct if infinite population – Approximately correct if population size N finite but much larger than sample size n • But if population is non-normal, what is the shape of the sampling distribution of the sample mean? – The sampling distribution is approximately normal if the sample is large enough, even if the population is non-normal (Central Limit Theorem) 7 -13

The Central Limit Theorem #2 • No matter what is the probability distribution that describes the population, if the sample size n is large enough, then the population of all possible sample means is approximately normal with mean mx=m and standard deviation sx=s/ n • Further, the larger the sample size n, the closer the sampling distribution of the sample mean is to being normal – In other words, the larger n, the better the approximation – norecord 7 -14

The Central Limit Theorem #3 Random Sample (x 1, x 2, …, xn) X as n large Population Distribution (m, s) Sampling Distribution of Sample Mean (right-skewed) (nearly normal) 7 -15