Hypothesis Testing and the T Test

First: Lets Remember Z Scores So: you received a 75 on a test. How did you do? If I said the mean was 72 what do you know? What if I said the standard deviation was 20? Or 10?

Z Scores A Z score is just a metric to standardize any distribution We don’t know what a 75 is without the mean and standard deviation. But if we knew the Z score is 1.5, we know exactly what that score means now.

Sampling (One Data Point) Lets say we have a distribution with a mean of zero and std. dev. of 1. (That is, a Z distribution). If you sample ONE item from this distribution, what is the probability that it came between -1 and 1? More than 2?

Sampling Multiple Data Points What if we drew two items from this distribution (with replacement) and then took the mean of those two items? Three Items? 1,000 Items?

Sampling From a Z Distribution

Z Test

What does this all mean? So, say the Census Bureau says that the average age (mean) in the US is 31. (With a standard deviation of 5) I have sampled 20 people in my neighborhood and their average (mean) age is 33. Is 33 a plausible average if my sample is typical of the US?

Hypothesis Tests The Null Hypothesis: This is the hypothesis that we are looking to disprove Usually, that there is “No Difference” i.e. My sample is the same as the population (in the Z test) In statistics the Null Hypothesis takes the form of the distribution of results that we would expect by chance More Likely Outcomes Less Likely Outcomes

Hypothesis Tests Remember, we have to take the upside down logic of how we would normally think about these things. We say, if the null hypothesis were true, is my sample probable? More Likely Outcomes Less Likely Outcomes

T Test for Independent Samples The question we are asking: I have two samples: sample A and sample B measured on some continuous variable Are these two samples from the same population? Assumptions: Independence (these are just random samples from a group) These are drawn from normally distributed populations with the same standard deviation

T – Test: What We’re Asking Do these samples come from the same population or different ones? So: lets say they came from the same population- could we expect these samples by chance?

T Test (Independent Samples)

More Hypothesis Testing What is the problem with asking questions like this: “Do these samples probably come from the same population?” Sometime we get unlikely results by chance Or the opposite: sometimes by chance, different populations will produce similar samples

Type I Error When we say that these two groups are probably from two different populations (but they are really from the same population) Formally: “Incorrectly rejecting the null hypothesis” i.e. the null hypothesis is true We set our probability of this error. It is called alpha. Generally we set it to.05.

Type II Error This is when we say that two samples are from the same population, but they are really from different population Formally, “Incorrectly accepting the null hypothesis, when the null hypothesis is false” This error we do not set, it depends on the size of the effect you hope to detect. Called Beta.