This test. Consider our respiratory disturbance index example again. A reasonable strategy, would be to reject the null hypothesis of our sample mean respiratory disturbance index was larger than some constant. Let's label that constant C. C will take into account the variability of X bar. Typically, C is chosen so that the probability of a type one error rate, this probability label is a low number. 5% has emerged as sort of a benchmark in hypothesis testing. So to repeat, alpha, is the type one error rate. Which in other words is, the probability of rejecting the null hypothesis when, in fact, the null hypothesis is correct. That's a bad thing, you don't want to make these kind of mistakes. But as in our court of law example, you don't want to set this rate too low, because then we would never reject a null hypothesis. Let's see if we can choose this constant C, so that the probability we would reject is simply tolerably low, say 5%. The standard error of the mean is 10, the assumed standard deviation of the population. And here we haven't drawn a distinction as to whether we estimate, or this is just a number that I've given you. Divided by square root 100, that's the square root of the sample size, that works out to be 1. Here I just created the settings, so it completely worked out to be 1. Under the null hypothesis where under H not mute is equal to 30, the distribution of the sample mean X bar is normal with a mean of 30, and a variance of one which we just calculated as the standard error square of the standard error of the mean in the line above. So, we want to choose the constant C. So that the probability that X bar is larger than C, under the null hypothesis is 5%. So remember the 95th percentile of the standard normal distribution is 1.645 standard deviations from the mean. So, if we set the constant as 1.645 standard deviations, from the mean under the null hypothesis. We will have achieved a cup point, so that the probability that a randomly drawn mean from this population is larger than this is 5%. So in this case it's 30, the hypothesized mean under the null hypothesis, plus 1, the standard error of the mean times 1.645, the number of standard deviations from the mean that we're considering which in this case works out to be 31.645. So, just to reiterate, the probability that a normal 30 with a mean of 30, and a variance of 1, is larger than this constant is 5%. So the rule, reject the null hypothesis when you receive an average larger than 31.645 has the property. That we will reject 5% of the time when the null hypothesis is true. Again, 5% of the time in the instances where the sample size is exactly a 100, and the standard deviation of the population is exactly 10. In the previous slide, we reverted the calculation of the rejection region C back to the original units of the data. However, I hope you got the gist from the problem that basically, whenever you are testing greater than, if the sample mean is more than 1.645 standard errors from the mean, from the hypothesized mean, you would reject. And there is nothing particular about 30, and the standard error of the mean of 1. So, instead of calculating this constant back on units of the original data, we tend to convert our sample mean into however many standard errors from the hypothesized mean it is. So, take this specific example. If our observed sample mean was 32, our hypothesized mean is 30, and our standard error is 10 divided by square root of 100, we're in a real. Problem if 10 would be estimated from the data. So, be the sample standard deviation, this works out to be 2. This is greater than 1.645 the chance of this occurring is less than 5%. So we're going to reject the null hypothesis. I, I should reiterate that the chance of this occurring, under the null hypothesis is less than 5%. So, we're going to reject the null hypothesis, in favor of the alternative hypothesis. So, I've just simply written out this rule again here on the final line. We're going to reject whenever X bar minus the hypothesized mean, divided by the standard error of the mean, is grater than the appropriate upper quantile that leaves Alpha percent in the upper tail.
