Chapter 18 Sampling distribution models math2200

Sample proportion Kerry v.s. Bush in 2004 –A Gallup Poll 49% for Kerry –A Rasmussen Poll 45.9% for Kerry –Why the answers are different? Sample proportion estimates population proportion There is randomness due to sampling

Modeling the Distribution of Sample Proportions Imagine what would happen to the sample proportions if we were to actually draw many samples. What would the histogram of all the sample proportions look like? –The histogram of the sample proportions to center at the true proportion, p, in the population –The histogram is unimodal, symmetric, and centered at p. –A normal model?

Model Let X be the number of people voting for Bush in a sample of size n Then X has a binomial model, Binomial(n,p) –p: the proportion of people for Bush in the entire population When n is large, we can use normal approximation –Normal model with mean np and variance npq

Modeling sample proportion Sample proportion is X/n –Normal model with mean p and variance pq/n

Example Back to Kerry v.s. Bush –Assume that the population proportion voting for Kerry is 49% –X/n has a normal model with mean 0.49 and standard deviation (n=1000) –Then we know that both 49% and 45.9 % are reasonable to appear

Conditions Normal model is an approximation to the exact model –Use it only when n is large –For example, if n=2, then X/n=0,0.5 or 1 1.Randomization Condition: The sample should be a simple random sample of the population. 2.10% Condition: If sampling has not been made with replacement, then the sample size, n, must be no larger than 10% of the population. 3.Success/Failure Condition: The sample size has to be big enough so that both and are greater than 10.

A Sampling Distribution Model for a Proportion Before we observe the value of the sample proportion, it is a random variable that has a distribution due to sampling variations. –This distribution is called the sampling distribution model for sample proportions. –We never actually take repeated samples from the same population and make a histogram. We only imagine or simulate them. –Still, sampling distribution models are important because they act as a bridge from the real world of data to the imaginary model of the statistic and enable us to say something about the population when all we have is data from the real world.

An example 13% of the population is left-handed. A 200-seat school auditorium was built with 15 “leftie seats” In a class of n=90 students, what’s the probability that there will NOT be enough seats for the left-handed students? Let X be the number of left-handed students in the class We want to find P(X>15) = P(X/n>0.167)

Check the conditions –n is large enough –randomization –10% condition The population should have more than 900 students –Success/failure condition np=11.7>10, nq=78.3>10 Normal model for X/n –Mean = 0.13 –Sd = sqrt(pq/n) = P(X/n>0.167) =

Sample Mean Sample means tend to normal when n is large

Central limit theorem (CLT) If the observations are drawn –independently –from the same population (distribution) the sampling distribution of the sample mean becomes normal as the sample size increases. We do not need to know the population distribution.

CLT Suppose the population distribution has mean μand standard deviation σ The sample mean has mean μand standard deviation σ/sqrt(n) Let X1, …, Xn be n independently and identically distributed random variables –E(X1) = μ –Var(X1)= σ 2 Then as n increases, the distribution of (X1+…+Xn)/n tends to a normal model with mean μand standard deviation σ/sqrt(n)

The Fundamental Theorem of Statistics The Central Limit Theorem (CLT) The mean of a random sample has a sampling distribution whose shape can be approximated by a Normal model. The larger the sample, the better the approximation will be.

Example Suppose the population distribution of adult weights has mean 175 pounds and sd 25 pounds –the shape is unknown An elevator has a weight limit of 10 persons or 2000 pounds What’s the probability that the 10 people who get on the elevator overload its weight limit?

Let Xi,i=1,2,…,10 be the weight of the ith person in the elevator Then we want to know P(X1+…+X10>2000) = From the CLT (check the requirement first), we know the distribution of is normal with mean 175 pounds and standard deviation Then

Standard error Using the CLT, we know the distribution of sample proportion is However, we do not know p in practice. Using the CLT, we know the distribution of sample mean is However, we do not know and

Standard Error When we don’t know p or σ, we’re stuck, right? Nope. We will use sample statistics to estimate these population parameters. Whenever we estimate the standard deviation of a sampling distribution, we call it a standard error.

Standard Error (cont.) For a sample proportion, the standard error is For the sample mean, the standard error is

The Process Going Into the Sampling Distribution Model

What Can Go Wrong? Don’t confuse the sampling distribution with the distribution of the sample. –When you take a sample, you look at the distribution of the values, usually with a histogram, and you may calculate summary statistics. –The sampling distribution is an imaginary collection of the values that a statistic might have taken for all random samples—the one you got and the ones you didn’t get.

What Can Go Wrong? (cont.) Beware of observations that are not independent. –The CLT depends crucially on the assumption of independence. –You can’t check this with your data—you have to think about how the data were gathered. Watch out for small samples from skewed populations. –The more skewed the distribution, the larger the sample size we need for the CLT to work.

Summary Sample proportions or sample means are statistics –They are random because samples vary –Their distribution can be approximated by normal using the CLT Be aware of when the CLT can be used –n is large –If the population distribution is not symmetric, a much larger n is needed The CLT is about the distribution of the sample mean, not the sample itself