# 1 Hypothesis Testing Chapter 8 of Howell How do we know when we can generalize our research findings? External validity must be good must have statistical.

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1 Hypothesis Testing Chapter 8 of Howell How do we know when we can generalize our research findings? External validity must be good must have statistical significance We need a way to ensure that the effects we find actually exist in the population, and not just in the sample Hypothesis testing does this

2 The need for hypothesis testing Imagine a study to check relationship between smoking and cancer We find that, in our sample, people who smoke more have a higher probability of getting lung cancer Can we believe that result applies to the population? No - might have been a fluke (all happened to get cancer due to something else)

3 Random variation Sometimes odd things happen to us Your car breaks in the middle of the road, your cellphone runs out of batteries and it starts to rain all at once Some things happen for a reason, some happen by just luck We tend to think that there is a reason for everything, but random variation can “cause” things to happen

4 Determining if random variation played a part The chance of my car breaking is about 10% could reasonably happen The chance of my phone dying is about 5% might reasonably happen The odds of it raining are about 10% could happen BUT the odds of it all happening at the same time is 0.05% (ie 1 in 2000)

5 More random variation If something has a 1 in 2000 chance of happening, and it does, it probably isn’t a fluke The odds are so small, it is more reasonable to find another explanation The same principle applies to testing scientific hypotheses What are the odds that the event we witnessed in our sample could have happened due to random variation?

6 The hypothesis test The general strategy: 1. Work out the statistic (correlation, etc) 2. Consider the conditions we collected our sample under (how many people, etc) 3. Calculate the probability that, given those conditions, that statistic could have occurred by random variation 4. If the odds are low, we reject the notion that random variation “caused” it

7 How to work out p for a hypothesis test In a hypothesis test, p is the probability that random variation “caused” the event Generally the same principle as for working out p for z Each statistic has a sampling distribution associated with it We use this to work out p, same way we used the z distribution to work out p.

8 The Null Hypothesis Each hypothesis test tries to show that random variation did not cause the event Shorthand of “random variation caused the event” is Ho - the Null hypothesis says, “nothing actually happened” The aim of a hypothesis test is to decide if Ho is so unlikely that we should reject it as an explanation for our results p is actually “probability that Ho is true”

9 Degrees of freedom p depends on a few things Sample size Number of groups in the design We express these conditions by calculating the degress of freedom of the statistic Often very simple, eg. df = n-1

10 How unlikely is unlikely enough? We need to decide how low a p value we are going to accept How low does the probability of random variation have to be for us to reject that idea? That level is called alpha (  ) it is usually set at 0.05 (5% chance) “if the chance that random variation caused this event are less than alpha (5%), then it must have been something else”

11 Working out p Awesomely difficult maths Get a computer to do it for you Once you have the p value, compare it to your alpha value if p is less than alpha then Ho is false (random variation did not play a part) if p is more than or equal to alpha, Ho is true (random variation might have played a part - can’t rule it out)

12 Hypothesis testing: step by step Step 1: decide on alpha, set out Ho alpha is normally 0.05 Ho is different for each stat, same basic idea “nothing happened” Step 2: work out the stat, work out p Step 3: compare p to alpha, decide if Ho should be rejected if p is less than alpha, reject Ho

13 Worked example: results from a computer You do a correlation, and get the following analysis: r = -0.3p = 0.04alpha = 0.05 Are these results statistically significant?

14 Worked example The computer has done steps 1 and 2 for you (you have the stat and p) You just do the step 3 - the decision (is p < alpha?) p (0.04) < alpha (0.05), so reject Ho - the results is statistically significant (ie. occurs in the population, not just the sample)

15 What have we discovered? What does it mean to find statistical significance? The relationship is likely to occur in the population - that it was not a fluke Tells us nothing about whether the relationship is positive or negative, or how strong it is How strong the relationship is, is expressed by the effet size - a different concept.

16 Error in hypothesis testing No hypothesis test gives a 100% sure result Out alpha sets our level of “confidence” - 0.05 means we are 5% sure we will make a mistake 2 kinds of of mistakes you can make: Type I error: you say Ho is false, when it’s actually true Type II error: you say Ho is true, when it’s actually false

17 Errors: Example Imagine you are a judge, hearing a case. You are presented the evidence If the guy did it, you must declare him “guilty” If he didn’t, you must declare him “not-guilty” Declaring someone who didn’t do it “guilty” is also a mistake (innocent man in jail) Declaring someone who did it “not-guilty” is a mistake!(the guy gets off)

18 Example Same idea in hypothesis testing You are presented the data If the Ho is false, you must reject it If the Ho is true, you must not reject it Rejecting a true Ho is a mistake Not rejecting a false Ho is also a mistake

19 Error types again Type I error is “putting an innocent man in prison” Type II error is “the crook getting away with it” We are interested in the probability of making one of these errors Cannot be avoided, only reduced Probability of making a Type I error is alpha

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