Chapter 18 Sampling distribution models math2200

Sample proportion Kerry vs. Bush in 2004 A Gallup Poll 49% for Kerry 1016 respondents A Rasmussen Poll 45.9% for Kerry 1000 respondents Why the answers are different? Variability of sample Kerry vs. Bush in 2004 A Gallup Poll 49% for Kerry A Rasmussen Poll 45.9% for Kerry Was one poll “wrong” ? Should we be surprised to find that we could get very different proportions from properly selected random samples Why the answers are different? Sample proportion estimates population proportion There is randomness due to sampling Last week we looked the example of 2004 presidential election poll. We were interested in finding the proportion of population in favor of Kerry. Two polls A Gallup poll showed 49% and a Rasmussen Poll a week later says 45.9%. Due to the variability of random sampling, every time we draw samples, we have something different.

Model Let Y be the number of people favoring Kerry in a sample of size n=1000 Y ~ Binomial(n,p) p: the proportion of people for Kerry in the entire population When n is large, Y can be approximated by Normal model with mean np and variance npq. Let p be the proportion of people favoring Kerry in the entire population. That’s is to say, the probability that a randomly selected respondent saying he would vote for Kerry is p. We may think the process of asking a randomly selected respondent as a Bernoulli trial and the probability that This person says he/she would vote for Kerry is p. Therefore we can model the total number of respondents who said yes out of 1000 by a binomial random variable. Then we used a computer software to simulate repeated samples and look at a histogram of the sample proportions of 2000 samples of size 1000 And noticed that the histogram looks like a normal model.

Modeling sample proportion The sample proportion Normal model with mean p and variance We figured out the mean and sd of the sample proportions and use the normal model with the same mean and the same sd to approximate the Distribution of the sample proportions.

Kerry vs. Bush (cont’) Assume the true population proportion voting for Kerry is 49%. The sample proportion = Y/n has a normal model with mean 0.49 and standard deviation 0.0158 (n=1000) Then we know that both 49% and 45.9 % are reasonable to appear (0.459 - 0.49)/0.0158= - 1.962

Sampling Distribution Model Consider the sample proportion as a random variable instead of a number. The distribution of the sample proportion is called the sampling distribution model for the 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 statistics and enable us to say something about the population when all we have is data from the real world. Sampling distribution models. They are used to model the sample to sample variations of statistics. For example, we used a Normal model to describe the sample proportion of people favoring Kerry.

Left-Handed: 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 Y be the number of left-handed students in the class. We want to find P(Y>15) = P(Y/n>0.167) = P( >0.167)

Left-Handed (cont’) 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 Approximate by Normal model for Y/n Mean = 0.13 = p Standard deviation = = 0.035 P( = Y/n >0.167) =normalcdf(0.167, 1E99, 0.13, 0.035) = 0.1446

Example: Sampling Distribution of a Mean 10,000 simulations for each graph.

Central limit theorem (CLT) If the observations are drawn independently from the same population (equivalently, distribution) the sampling distribution of the sample mean becomes normal as the sample size increases. The population distribution could be unknown.

CLT Suppose the population distribution has mean μand standard deviation σ The sample mean has mean μand standard deviation . Let Y1, …, Yn be n independently and identically distributed random variables E(Y1) = μ Var(Y1)= σ2 Then as n increases, the distribution of (Y1+…+Yn)/n tends to a normal model with mean μand standard deviation The more general result theCLT says if our sample values are independent and as the sample increases, the sample mean becomes a normal r.v. That’s to say if n independent r.v from the same distribution with mean mu and sd sigma, the mean of these r.v follow a normal Distribution with mean mu and sd sigma/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: Elevator Overloaded 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, the distribution of sample proportion is In general, by the CLT the distribution of sample mean of independent sample values is p, and could be unknown in some cases.

Standard Error If we don’t know or σ, the population parameters, we will use sample statistics to estimate. The estimated standard deviation of a sampling distribution is called a standard error.

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

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