Sampling Distributions Welcome to inference!!!! Chapter 9

Parameter  A Parameter is a number that describes the population.  A parameter always exists but in practice we rarely know it’s value b/c of the difficulty in creating a census.  We use Greek letters to describe them (likeμor σ). If we are talking about a percentage parameter, we use rho (ρ)  Ex: If we wanted to compare the IQ’s of all American and Asian males it would be impossible, but it’s important to realize that μ Americans and μ males exist.  Ex: If we were interested in whether there is a greater percentage of women who eat broccoli than men, we want to know whether ρ women > ρ men

Statistic  A statistic is a number that describes a sample. The value of a statistic can always be found when we take a sample. It’s important to realize that a statistic can change from sample to sample.  Statistics use variables like x bar, s, and phat (non greek).  We often use statistics to estimate an unknown parameter.  Ex: I take a random sample of 500 American males and find their IQ’s. We find that x bar =  I take a random sample of 200 women and find that 40 like broccoli. Then phat w =.2  IMPORTANT! A POPULATION NEEDS TO BE AT LEAST 10 TIMES AS BIG AS A SAMPLE TAKEN FROM IT. IF NOT, YOU NEED A SMALLER SAMPLE

Bias  We say something is biased if it is a poor predictor

Variability *Variability of population doesn’t change- (scoop example) size of scoop matters

How can we use samples to find parameters if they give us different results?  Imagine an archer shooting many arrows at a target: 4 situations can occur  a) consistent but off target. b) all over the place. tends to average a bulls eye but each result is far from center. c) worse than a as the archer is consistently missing high and to the right but nearly as consistently as situation a. d) is ideal- low bias and low variability.

Here’s an example:  Suppose our goal was to estimate μ american male.  We can’t take a census, so we take many samples. We find the average IQ of american males in each sample (x bar).  a) if our many samples of IQ’s are consistent but higher than the true average IQ of AM’s, then we have a situation with high bias and low variability  b) If our many samples are inconsistent- some high, some low than the true mean of AM’s Iqs, we have low bias and high variability.

 If our sample means are not close to each other but all higher than μ AM then we have high bias and high variability (c)  Finally, if our samples all just slightly higher or slightly lower than μ AM, we have our desired situation: low bias and low variability.

But…  You aren’t taking a bunch of samples…you’re only going to take 1! and we want it to predict μ am  If we used the data from situation d) then any of the samples would provide a good predictor for μ am  We already know some ways to get a good sample- using an SRS and being very sure to have no bias when choosing our sample.  Inference is using our sample statistic (assuming it’s a good sample) to predict our parameter with a certain degree of confidence.

The Sampling Distribution  The sampling distribution of a statistic is the distribution of means of all possible samples of the same size from the population.  When we sample, we sample with replacement.  A sampling distribution is a sample space- it describes everything that can happen when we sample. Cool demo

Central Limit Theorem (CLT)  As you take more and more SRS’s of the same size, the distribution of their means will get closer and closer to a normal curve centered around the true population mean…NO MATTER WHAT THE SHAPE OF THE PARENT POPULATION!! Why?!  The Sampling Distribution of means has a mean of μand a standard deviation of σ/√n** N(μ, σ/√n)

CLT summary  1. The mean of the population (what we want to find) will be the same as the mean of all your many samples.  2. The Standard Deviation of all your many samples will be the population standard deviation divided by √n (your sample size)  3. The histogram of the samples will appear normal (bell shaped).  4. The larger the sample size (n), the smaller the standard deviation will be and the more constricted the graph will be.

What if we’re talking about proportions?  Same thing except we use our proportion formulas for mean and standard deviation

Don’t forget our rule of 10!  Only use σ/√n or if the 10 times N (the number of people in our population) is ≥ n (the number of people in our sample)

Example  The true average study time for a final exam in history is found to be 6 hours and 25 minutes with a standard deviation of 1 hour and 45 minutes. Assume the distribution is normal. N(6.417, 1.75)  What is the probability that a student chosen at random spends more than 7 hours studying? Normalcdf(7,100,6.417,1.75) = 37%  What is the probability that an SRS of 4 students will average more than 7 hours in studying? Normalcdf(7,100,6.417,1.75/√4) = 25.3%.  Why did the probability go down?  A student to study more than 7 hours is not probable…a group of 4 to average more than 7 is less probable.

Example 2  The length of pregnancy from conception to birth varies normally with a mean of 266 days and a standard deviation of 16 days  N(266,16)  What is the probability that a woman chosen at random has a pregnancy lasting more than 270 days?  40.1%  What is the mean and standard deviation of my sampling distribution?  What is the probability that an SRS of 16 women have pregnancies averaging more than 270 days?