Statistical Inference & CI's Confidence Intervals

Statistical Inference Drawing conclusions about the population from sample data Use probability to express strength of conclusions Statistical Inference assumes your data comes from a simple random sample or randomized experiment!!! Keep this idea in mind for your BIG project…

Remember this? IF your data is normally distributed… 68% of the data lies within 1 standard deviation of the mean. 95% of the data lies within 2 standard deviations of the mean 99.7% of the data lies within 3 standard deviations of the mean

Little Lost Gargamel wants to know the true mean of Smurf BMI’s. Smurf BMI: m = ? s = 13.72 Little Lost Gargamel wants to know the true mean of Smurf BMI’s. He takes a sample of size 30. His sample mean = 21.6 He asks you to give him an interval where you are pretty certain the whole smurf population falls…

Little Lost We know that is normally distributed thanks to the CENTRAL LIMIT THEOREM! Smurf BMI: m = ? s = 13.72 I’m a sample mean. 68% of us are WITHIN a s Where are you m?? 24.05? Where are you m?? 19.15? So I’m going to go hunting for m Where is my m? 2.45 2.45 Remember that 68% of the results should be within 1 std dev of the mean…

I am 68% certain that the population mean is between 19.15 and 24.05. Watch Your Wording!!!! 68% of the intervals constructed with this method would contain the population mean. I am 68% certain that the population mean is between 19.15 and 24.05. Let’s also look at how NOT to word it…

68% of the time the population mean will be between 19.15 and 24.05 Watch Your Wording!!!! 68% of the time the population mean will be between 19.15 and 24.05 There is a 68% chance that the population mean is between 19.15 and 24.05 There is a 68% probability that the population mean is between 19.15 and 24.05 So How do I Make a Confidence Interval?

The Formula Standard Error How to get LESS margin of error You find these in the t-table in the back cover of your book. Check out the whiteboard for some sweet math action to find other z*’s… This depends on how confident you want to be How to get LESS margin of error Decrease confidence level Increase sample size

What's the Secret Ingredient? I know the standard deviation of secret ingredient mix in the Original Recipe Chicken Bucket is 3.5 oz. I want you to be 99% confident and tell me what you think the overall mean ounceage is… In the sample of 100 buckets you asked me to look at, I found a mean of 18.2 ounces… Does that help?

17.2984 – 19.1016 σ = 3.5 and n = 100 So let’s look at what we know… x-bar = 18.2 z* = ? Z* = 2.576 So let’s look at what we know… Now, let’s get that interval… Looks like we need a z* for 99% Confidence… 17.2984 – 19.1016

What’d You Find Pardner? What's Your Report? Good Work, Geek!!! What’d You Find Pardner? Actually sir, What he meant to say was we’re 99% certain the mean ounceage of secret ingredient is between 17.2984 and 19.1016. Aargh!! What a Moron!! Well Sir, I believe there’s a 99% chance the population mean falls between 17.2984 – 19.1016 ounces per bucket…

Finding Sample Size for a Desired Margin of Error Use the formula below if you want to find the sample size you need to produce a specific margin of error A study of the career paths of hotel managers taken from an SRS of 116 found the average time they had spent with their current company was 11.78 years. With a known standard deviation of 3.2 years, how large of a sample would need to be taken to estimate the mean within one year with 99% confidence? We would need a sample size of at least 68. Where m is the margin of error you are looking for!!

Cautions for Inference Data must be from an SRS Inference has no correct method for data collected in a method different from SRS Outliers have a strong effect on CI’s Be sure to use the distribution of x-bar because of it’s normality For our current method, you MUST know σ

The Final and Biggest Warning!!! A Few Last Warnings The margin of error in a CI covers random sampling errors Difficulties from sample surveys and opinion polls are often not covered in the margin of error 95% Confidence DOES NOT say that there is a 95% probability that the true mean falls within the interval. The True Meaning of a CI The numbers were calculated by a method that gives correct results in 95% of all samples. (you can say we are 95% confident or certain the mean is within the interval) The Final and Biggest Warning!!!

Using the Calculator for CI’s Stat – Tests – 7: ZInterval - Stats Put in , x-bar, n, and C-Level *You can also use a list of stats (Data) Calculate A random sample of 60 employees of a large corporation had a sample mean of 32.5 vacation days with known standard deviation of 18.5 days. Find a 90% confidence interval for the population mean of accumulated vacation days.

Homework #9,10,13,18-26