Chapter 18 – Sampling Distribution Models

How accurate is our sample? Sometimes different polls show different results for the same question. Since each poll samples a different group of people, we should expect some variation in the results. We could try drawing lots of samples and looking at the variation amongst those samples.

Coin flipping experiment Open up the web page: http://shazam.econ.ubc.ca/flip/ Flip 10 coins and record the number of heads Flip 50 coins and record the number of heads Flip 100 coins and record the number of heads Let’s look at a histogram with your results

Sampling Distribution Model for a Proportion Our histogram of the sample proportions started to look like a Normal model The larger our sample size gets, the better the Normal model works Assumptions: Independence: sampled values must be independent of each other Sample Size: n must be large enough

Conditions to check for assumptions Randomization Condition: Experiments should have treatments randomly assigned Survey samples should be a simple random sample or representative, unbiased sample otherwise 10% Condition: Sample size n must be no more than 10% of population Success/Failure Condition: Sample size needs to be large enough to expect at least 10 successes and 10 failures

Sampling Distribution Model for a Proportion If the sampled values are independent and the sample size is large enough, The sampling distribution model of is modeled by a Normal model with:

Example based on our coin flips 10 coins 50 coins 100 coins Mean = SD =

Model of 100 coin flips

Example: Proportion of Vegetarians 7% of the US population is estimated to be vegetarian. If a random sample of 200 people resulted in 20 people reporting themselves as vegetarians, is this an unusually high proportion? Conditions: Randomization 10% condition Success/Failure

Vegetarians Example continued Since our conditions were met, it’s ok to use a Normal model. = 20/200 = .10 E( ) = p = .07 z = This result is within 2 sd’s of mean, so not unusual

68-95-99.7 Rule with Vegetarians 68% 95% 98% -3σ -2σ -1σ p 1σ 2σ 3σ

More with vegetarians What is the probability that we get a random sample of 200 people with 10% vegetarians?