Sampling Distributions & Sample Means Movie Clip

Why Sample? A sample is a small portion of the population that is designed to give accurate results about the population Sampling saves a lot of time and money and is more realistic

Parameter vs. Statistic Parameter –Number that describes the population Often don’t know cuz we can’t examine a population (most of the time) Fixed # Statistic: –# that describes the sample. Found from taking a sample Changes from sample to sample Used to estimate unknown parameters Movie Clip

Parameter vs. Statistic Parameters –  - population mean –  - population standard deviation –p – population proportion Statistic: –x-bar – sample mean –s – sample standard deviation –p-hat – sample proportion These are either ideal #’s created by company or projected #’s… Not KNOWN for large populations !! These #’s are collected from samples. Samples taken should be the same size. Each sample can give different results… Mean of Means =

What’s the Problem with Sampling? Different Samples Give Different Results What’s Your Weight? Movie Clip

Sampling Variability We often use “statistics” to make inferences about the population So What’s the Problem? The value of a statistic varies from sample to sample when you are taking repeated random samples. The concept listed above is called Sampling Variability. How do we counteract this problem? We take a whole bunch o samples, hence sampling distributions.

Sampling Distribution Sampling Distribution of a Statistic –the distribution of values taken by the statistic in all possible samples of the same size from the same population –Ideal pattern that would be displayed if we looked at all possible samples of a population –Used to make inferences about parameters –Distribution of all collected sample means defined by a mean and standard deviation Movie Clip

Describe a Sampling Distribution Center –Description of μ xbar x-bar or p-hat vs. the median Shape –“often” normal, but looking for skewness Spread –Talk about range and standard deviation (std dev is especially strong descriptor for normal distributions), also look for outliers

Fighting Sampling Variability Below are 4 keys to fighting Sampling Variability –Take a large # of samples from the same population (all the same size) bigger samples are always better!! –Calculate the sample mean, x-bar, or the sample proportion, p-hat, for each sample –Make a histogram of the values of x-bar or p- hat –Examine the distribution displayed in the histogram for shape, center, and spread, as well as outliers and other deviations Sampling Distribution Movie Clip

Bias of Statistic Bias – Difference from Overall Sample Mean (from Sampling Distribution) and Unknown Population Mean –Relation of the statistic to the parameter –Unbiased Statistic: Mean of sampling distribution is equal to the true value of the parameter being estimated μ xbar = , p-hat = p Variability – difference in values from sample to sample –The spread of the sampling distribution Larger samples = Less variability(spread)

Bias vs. Variability Target Practice Low Variability, High Bias High Variability, Low Bias Low Variability, Low Bias Good Shootin’ Pardner!!! That’s the what we like to SEE!! If I was a sampling distribution, I’d want to be THAT ONE!!!! High Variability, High Bias

Sampling Distributions of Means (quantitative) μ xbar is unbiased estimator of µ μ xbar = µ Means are less variable than individual observations Means are MORE normal Standard Deviation of means is effected by the sample size Sample Size (n) Standard Deviation (σ) Sample Size (n)

Variation of Means 4 times sample size to ½ std dev Population should be 10 times size of sample –Formulas and characteristics may not be accurate if this condition is not met Movie Clip

Using Normality to Your Advantage Samples taken from Normal Distributions are normal… Allows us to use the normal curve to calculate the probability of your sample occurring EX: What is the probability that a randomly selected student would be shorter than 63 inches? What is the probability that an SRS of 15 students has a mean height less than 63 inches? Let’s Practice This From Our Height Worksheet … Smurf Worksheet

Central Limit Theorem What if it’s not normal?? Bigger sample size more normal distribution How big? The more out of wack, the bigger your samples need to become For your thoughts… Look at your Height Worksheet… How can you compare the shape of the distributions as you increased the sample size?

Homework Problem #’s 12-17,35-38 Bring the Coins Listed Below and Fill Out Part 1 on the Worksheet