0:07

A reasonable strategy, would be to reject the null hypothesis of our sample mean

Â respiratory disturbance index was larger than some constant.

Â Let's label that constant C.

Â C will take into account the variability of X bar.

Â 0:22

Typically, C is chosen so that the probability of a type one error rate,

Â this probability label is a low number.

Â 5% has emerged as sort of a benchmark in hypothesis testing.

Â So to repeat, alpha, is the type one error rate.

Â Which in other words is, the probability of rejecting the null hypothesis when,

Â in fact, the null hypothesis is correct.

Â That's a bad thing, you don't want to make these kind of mistakes.

Â But as in our court of law example, you don't want to set this rate too low,

Â 1:12

The standard error of the mean is 10,

Â the assumed standard deviation of the population.

Â And here we haven't drawn a distinction as to whether we estimate, or

Â this is just a number that I've given you.

Â 1:23

Divided by square root 100,

Â that's the square root of the sample size, that works out to be 1.

Â Here I just created the settings, so it completely worked out to be 1.

Â 1:34

Under the null hypothesis where under H not mute is equal to 30,

Â the distribution of the sample mean X bar is normal with a mean of 30,

Â and a variance of one which we just calculated as the standard error square of

Â the standard error of the mean in the line above.

Â 1:51

So, we want to choose the constant C.

Â So that the probability that X bar is larger than C,

Â under the null hypothesis is 5%.

Â So remember the 95th percentile of the standard normal distribution is

Â 1.645 standard deviations from the mean.

Â So, if we set the constant as 1.645 standard deviations,

Â from the mean under the null hypothesis.

Â We will have achieved a cup point, so that the probability that a randomly drawn

Â mean from this population is larger than this is 5%.

Â 2:42

So in this case it's 30, the hypothesized mean under the null hypothesis, plus 1,

Â the standard error of the mean times 1.645, the number of standard deviations

Â from the mean that we're considering which in this case works out to be 31.645.

Â So, just to reiterate, the probability that a normal 30 with a mean of 30,

Â and a variance of 1, is larger than this constant is 5%.

Â So the rule, reject the null hypothesis when

Â you receive an average larger than 31.645 has the property.

Â That we will reject 5% of the time when the null hypothesis is true.

Â Again, 5% of the time in the instances where the sample size is exactly a 100,

Â and the standard deviation of the population is exactly 10.

Â 3:35

In the previous slide, we reverted the calculation of the rejection region

Â C back to the original units of the data.

Â However, I hope you got the gist from the problem that basically,

Â whenever you are testing greater than, if the sample mean is more than

Â 1.645 standard errors from the mean, from the hypothesized mean, you would reject.

Â And there is nothing particular about 30, and the standard error of the mean of 1.

Â So, instead of

Â calculating this constant back on units of the original data, we tend to convert our

Â sample mean into however many standard errors from the hypothesized mean it is.

Â So, take this specific example.

Â If our observed sample mean was 32, our hypothesized mean is 30,

Â and our standard error is 10 divided by square root of 100, we're in a real.

Â Problem if 10 would be estimated from the data.

Â So, be the sample standard deviation, this works out to be 2.

Â This is greater than 1.645 the chance of this occurring is less than 5%.

Â So we're going to reject the null hypothesis.

Â I, I should reiterate that the chance of this occurring,

Â under the null hypothesis is less than 5%.

Â So, we're going to reject the null hypothesis,

Â in favor of the alternative hypothesis.

Â So, I've just simply written out this rule again here on the final line.

Â We're going to reject whenever X bar minus the hypothesized mean,

Â divided by the standard error of the mean, is grater than

Â the appropriate upper quantile that leaves Alpha percent in the upper tail.

Â