This video is about significance tests about a population mean. How long can scuba divers stay underwater? Well, that depends on the size of their oxygen tank, their experience, the depths of the dive and many more things. Suppose you have very good reasons to expect that experienced American divers who dive with an average tank at an average depth, will stay underwater for more than 60 minutes. Suppose you have also approached 100 experienced American scuba divers and measured how long they could stay underwater with an average tank at an average depths. You find that the mean time these divers spent underwater is 62 minutes. The standard deviation is five minutes. You expect that the divers will stay underwater for more than 60 minutes. That leads to the following null hypothesis. Mu equals 60. The alternative hypothesis is mu is larger than 60. We conduct a significance test about a population mean like this. We assess if it's likely that the sample we have collected, actually comes from a population with a mean that equals the value formulated in the null hypothesis. So this is the distribution we're interested in. It is the sampling distribution of the sampled mean, with a mean of 60, which is the null hypothesis value. How likely is a sampled mean of 62 if the population mean is 60? To answer that question, we compute a test statistic. That's the number of standard errors the sample mean is removed from the population mean according to the null hypotheses. You might remember that to compute the standard error, we need to know the value of the population standard deviation. But because we don't know that value, we have to estimate it using the sample standard deviation. Because this implies that we introduced extra error, we employ the t distribution instead of the z distribution. Our test statistic can be computed with the following formula. It is the sample mean minus the null hypothesis mean divided by the standard error of the sample mean.The standard error equals the sample standard deviation divided by the square root of the sample size. Let's first compute the standard error. That's 5 divided by the square root of 100, that's 0.5. That makes 62 minus 60 divided by 0.5 equals 4. Our test statistic has a t score of 4. Is that enough proof to reject the null hypothesis? Well, that depends on the significance level. Let's employ the most common significance level of alpha equals 0.05. We do a one-tail test, so this is where our rejection region is. The critical value here can be found in a t table, it is 1.67. Note that we look at 60 degrees of freedom because we have 99 degrees of freedom, and 60 is the closest lower value in the table. We look at t 90% because we want a cumulative probability of 0.05 in the right tail of the distribution. You need to remember that t 90% score stands for the confidence level of 90%, which assumes that we have 10% in both tails of the distribution together. That means that we have to take into account that, there's also 0.05 in the left tail. This is the result. Our test statistic falls within the rejection region, which means that we reject the null hypothesis that the population mean is 60 minutes. We can conclude that on average, experienced American divers with an average tank at an average depth stay underwater for more than 60 minutes. What if our expectation is not that experienced divers who dive with an average tank at an average depth stay underwater more than 60 minutes, but that they stay underwater either more or less than 60 minutes. Our alternative hypothesis then becomes mu is not 60. Now, we have to do a two-tailed test. Say we set the significance level at 0.01. Our sampling distribution then looks like this. We have a cumulative probability of 0.005 here and here. If we look up the critical values we find -2.66 and 2.66. Our test statistic is 4. So we still reject the null hypothesis and conclude that our finding is statistically significant. Because we did a two tail test our substantive conclusion now is that the mean time experienced American divers with an average tank at an average depth stay underwater is not 60 minutes.