 So one of the more important situations where we test a hypothesis is when we have normally distributed outcomes. And that's because, given an observed event, it can be very tedious to compute the probability of all events of equal or lesser probability. But if the underlying probability distribution is normal, the problem is much easier. And that's because the probability of the observed event is the height of the normal curve at the corresponding point, and all lower probability events are further away from the mean on both sides. For example, suppose you're testing light bulbs which are supposed to have a lifetime that is normally distributed with mean equal 10,000 hours and a standard deviation of 500 hours. A light bulb burns out after 90-200 hours. Let's find the p-value corresponding to this observation and interpret its significance. Now the first thing we'd like to do here is determine how far away from the mean we actually are. And so the thing to start with is that 90-200 is 800 hours below the mean. That's 10,000 minus 90-200. And that means the events of equal or lower probability will be that a light bulb burns out after less than 10,000 minus 800, 90-200 hours. Or a light bulb burns out after more than 10,000 plus 800, 10,800 hours. So if the null hypothesis is true, then the probability of an event with equal or lower probability will be the probability that our light bulb burned out in less than or equal to 90-200 hours. Or it lasted greater than or equal to 10,800 hours. Now if a light bulb burns out after less than or equal to 90-200 hours, it certainly won't last more than or equal to 10,800 hours. So these events are mutually exclusive and so we can split the probabilities. We'll have the probability the light bulb last less than or equal to 90-200 hours. Plus the probability the light bulb last more than or equal to 10,800 hours. So using the normal distribution with mean 10,000 and standard deviation 500 and our calculator, computing device, or random pass or by, we find the probability the light bulb last less than or equal to 90-200 hours is 0.0548. And because our interval is symmetric about the mean, then the probability our light bulb last longer than or equal to 10,800 hours is going to be the same thing. Substituting these in, we find that the probability of observing an event with equal or lower probability is 0.1096. That's about 11% or roughly speaking one chance in 9. So it seems reasonably likely that light bulbs with a mean lifetime of 10,000 hours and standard deviation of 500 hours could yield such an observation. Well we still have to decide whether to reject or fail to reject the null hypothesis. And so the thing to remember is that the p-value corresponds to the probability of observing the outcome if the null hypothesis represents the true state of the world. It should inform our decision but we still have to make a decision and in order to do that we need to consider the consequences if we're wrong. So let's add a little context. Remember this context does not come from mathematics but it comes from what you know about how the real world operates. So suppose we know that rejecting the null hypothesis and coming to the conclusion that the light bulbs do not have a lifespan of 10,000 hours requires rebuilding the assembly plant costing hundreds of millions of dollars and laying off thousands of workers. On the other hand if the light bulbs don't have the specified lifespan the company may be fined a million dollars. How would you decide if you were part of the company? So it's helpful to break down the decisions in terms of their consequences and whether the decision was correct. So again either the null hypothesis is the true state of the world or it isn't and either we reject the null hypothesis or we fail to reject it. If the null hypothesis is true and we've rejected it that's an error otherwise it's a correct decision. If the null hypothesis is not true and we fail to reject it that's an error otherwise we've made the correct decision. And with the context if we reject the null hypothesis whether or not it's true we'll have cost the company hundreds of millions of dollars and lost a bunch of jobs. On the other hand if we fail to reject the null hypothesis the company could be subject to a million dollar fine. And so a rational sensible logical person would take a look at the consequences. And so we might decide the consequences of rejecting the null hypothesis are worse than the consequences of not rejecting the null hypothesis even if that was the correct decision. So unless we have strong evidence against it we should not reject the null hypothesis. Now an important idea here to keep in mind is that consequences are personal. The decision you make depends on who you are. So if you're a buyer of these light bulbs then rejecting the null hypothesis will mean you'll have to find a new light bulb supplier. But if the light bulbs don't have the advertised lifespan you will need to buy and replace light bulbs more often. So how do we decide now? So again let's break this down into whether our decision is correct or not and what the consequences are. So here rejecting the null hypothesis means we have to find a new supplier while failing to reject the null hypothesis means you have to buy more and replace more light bulbs. Now finding a new supplier means we have to go to a different store or buy a different brand. Whereas buying more and replacing more light bulbs means a lot of effort for us. So here you might decide it's more of a hardship to have to buy and replace new light bulbs than it is to find a new supplier. So we might be more willing to reject the null hypothesis on weaker evidence. Or let's take another example completely hypothetical of course. Ten thousand scientists has to hypothesis. Rejecting the null hypothesis will cost billions of dollars. But if the null hypothesis is true failing to reject it will lead to the extinction of humanity. How should you decide if you're a member of humanity? So again we should consider the consequences. Rejecting the null hypothesis is going to cost billions. Failing to reject the null hypothesis could lead to human extinction. If we reject the null hypothesis then even if it was the right thing to do it'll cost billions of dollars. But failure to reject the null hypothesis risks human extinction. Well any rational sensible person will say billions of dollars that's terrible. We should risk human extinction. Wait, wait wrong script. Any sensible rational person is going to decide that the consequences of rejecting the null hypothesis are milder than the consequences of failing to reject the null hypothesis. And so we should be inclined to reject the null hypothesis even if the evidence is weaker. So even if only some of the scientists reject the null hypothesis we should still consider rejecting the null hypothesis. And if almost all of the scientists reject the null hypothesis that's strong evidence. And so we really ought to reject the null hypothesis.