Singing So we’’re gonna do a hypothesis test about whether an average speed of a dog is less than or equal to thirty miles per hour and the alternative being greater than thirty. Now the goal of this video’’s to talk about What is a p-value. What’’s the definition. What does it really mean? And the answer, it means exactly what we practiced all last chapter Those probabilities for mean So. If We’’re gonna do the definition of P-value there’’s three pieces to it, Number one says “under the null hypothesis, Assuming the null hypothesis is true”. In other words, when I draw my picture, I’’m gonna draw it over. Whatever the null hypothesis said it would equal. Here it says it would equal thirty. So thirty is what goes here. The second part says “What’’s the probability of the statistic” So. Let’’s assume our statistic was 32 miles per hour. And so we’’re gonna say what’’s the probability of 32 But if we said what’s the probability of exactly 32 there’s no area there We’d have a probability of zero. It needs to say in what direction? That’’s why the third step is “probability of that or more extreme”. Now what “more extreme” means depends on this alternative. So over here, if our alternative says greater than more extreme means what the probability of that and greater The less than means you would want the area that is the statistic or less. The hard part is when it’’s not equals. Because that means you want the statistic or further away from mean. So our example, the way we have it right now, Our alternative says greater than thirty. So our p-value is going to be “What’’s the probability of 32 or more” Here with this alternative. Our p-value, with the z-score of 2, would be .02275. So when I say the P-value is .022, what I’’m trying to say is “If. The mean really was thirty. Then the probability of getting thirty or more is .02275. Let’’s change our alternative like this. Now we want the area to the left. So instead of saying “What’’s, the probability of getting 32 or more” I’m gonna say “What’’s the probability of getting 32 or less”. Which means I need to flip my side here and there’’s my p-value. Now .97725. My p-value completely changed Because the question I was asking completely changed. I want to know is the average less than thirty. And my data says “woah No way” It’s so not less than thirty it’’s, possibly greater than thirty. And your p-value’’s gonna be huge to say “Look. If the average is thirty or more than this, kinda data would be very likely”. So do we have evidence to support? The average is less than thirty. No, no! The evidence is pointing towards the null hypothesis. So we get a huge p-value. Let’’s try the hard scenario. Test whether the mu is equals thirty or not equals thirty. So our question now is “Could Mu be far from thirty in either Direction”. I don’t know if it’’s too far up or too far down, but both directions matter Now, just by luck, we got on the high end, but the null hypothesis says. Come on, if you redid this study. You’’d probably end up on the low end or somewhere that far away. So, for the p-value we don’’t just want to say “What’’s the probability of being even further on the high end” we want to say, also “What’s the probability of being that far out on the low end”, In other words, the probability of being that far away from thirty in either direction. Now the applet doesn’’t do a two tailed are like that And it’’s partly on purpose, partly because I think it would be confusing. So what we’’re gonna do is we’’re gonna say “I need this tail, and I know I have that much as the same area on the flip side”. So this much area is .02275. I know this is gonna be the same area .02275. And now my p-value is gonna be a combination of both those. Which is .0455. Here’’s some very common questions. Students ask. “how, do I know I’’m supposed to be doubling my p-value”? The answer is if it’’s a not equals than you need to find the small tail and double that to get your p-value. “what does the p-value mean” Well? If the null hypothesis was true, we would see this kinda data only two percent of the time. We set our cutoff here to be .05. So this happens more rare than what we’’d call rare, which is. Our alpha .05. So something is weird. Something’’s wrong. Our data shouldn’’t happen. It was very unlikely. We’’re gonna declare the null must be wrong. Now, in this case, with the null hypothesis, the probability of getting data this high or something lower than that 97% chance, which is a lot of support in favor of the null. So a high probability here. 97% chance data. It looks like our null hypothesis was doing fine. If this was how we set up our null Here, what’’s the chance of getting data this far away from the null or more well, there’’s only a 4% chance. Still less than our five percent. So this data was unlikely, according to the null hypothesis, Which makes us think. Maybe the null hypothesis was wrong. Calculating the p-value’’s the same math as you’’ve been doing before. The only thing we’’re doing now is using that in interpretation to come to a conclusion. Your p-value is evidence supporting the null Lot of evidence to support the null…, not a lot of evidence to support the null. Based on the p-value, we can decide “Should we reject or fail to reject our null hypothesis and make conclusions”.