Welcome to my second video on hypothesis Testing in this example, we’re going to perform a one-tailed test with a t-distribution. So let’s get started here. In this example, it says the average. IQ of the adult population is 100 A researcher believes that the average IQ of adults is lower. A random sample of 5 adults are tested and the scores are given below. Is there enough evidence to suggest that the average? IQ is lower So although this example, is different than my first video I made on hypothesis testing. These steps are going to remain exactly the same, so let’s go over all of the steps to perform a hypothesis test so here. I have written for you. All of these steps and step number One says to state the null, which is written with H sub 0 or H naught and the alternative hypothesis and the alternative hypothesis is written with H Sub 1 or H Sub. A it doesn’t matter which one you use, so we have to state our null and alternative hypothesis, So let’s go back to our example, and let’s start with our null hypothesis. H naught, we’re going to state our null hypothesis. Now the null hypothesis is what is currently believed to be true or what is currently accepted. So in this example, it says the average. IQ of the adult population is 100 so the currently accepted value for the average IQ of adults is 100 so the average of the population is equal to 100 This is our null hypothesis, and the null hypothesis is always written with an equal sign. So that’s just a little tip. You’re always going to use an equal sign for the null hypothesis, So now lets. State our alternative hypothesis. Our alternative hypothesis is written with H Sub 1 or H sub. A It doesn’t matter and the alternative hypothesis is always what is being claimed. In this example, it says a researcher believes he is claiming that the average. IQ of adults is lower and these two words are really important is lower. He’s claiming that the average has gone down is decreased because it is lower, so our alternative hypothesis is that the average which is written with this. Greek letter Mu. The average is lower than 100 it has decreased so our alternative hypothesis is that the average is less than 100 and anytime that the alternative hypothesis is written with a less than or greater than symbol, This means we’re going to perform a one tailed test. I’m going to explain what this means in my next step, but anytime you use a greater than or less than symbol in your alternative hypothesis, This means we’re going to perform a one tailed test so now we are ready to move on to our next step. So if we go back to our hypothesis, Testing steps, step number two says to choose the level of significance, which is written with the Greek letter Alpha. Now this is just the area in the tail, so let’s go back to our example. We need to choose our level of significance, which is just the area in the tail. So let me draw a picture of the normal curve for you just so. I’ll show you what this looks like, so like I said before. The level of significance is just the area in the tails and the reason why I chose a level of significance of point. Zero Five is because this just seems to be the most common value used and this problem didn’t give us the level of significance to use, so I just chose the most common value of point zero five and notice how the only area or the only tail that I shaded in the area was the bottom tail. This is because we’re performing a one tailed test and the reason why I chose to use. The bottom tail in the upper tail was because the researcher is claiming that the average IQ is lower. He’s claiming that it’s lower so the only area of interest is below the average, So if you draw the average of the population on the normal curve, it’s always in the middle so here in the middle, we have our average of the population and in the researcher is claiming that this average has lowered it’s decreased. So the only tale of interest is our lower tail. This is a one tailed test and once again. Our level of significance is point Zero Five. So we know the area inside this tail that I shaded in red. We know that this area is equal to point zero five. So now that we’ve stated our level of significance, let’s move on to step number three. Now, step number three says to find the critical values now. The critical values are just the Z value or the T value that separates the tail from the rest of the curve now. I mentioned earlier that we’re using a t-test or T distribution for this particular example, so our critical value is going to be a T value. Now, let me explain to you. Why we’re using a t-test for this particular example, and I wrote the two conditions when we need to use a T value and condition number one says the population standard deviation, which is Sigma is unknown. Condition number two says that the sample size is less than 30 So if we go back to our example, let’s see if these two conditions have been met, it says the average. IQ of the adult population is 100 but it never gives us the population standard deviation. The only thing it gives us is the standard deviation of the sample. So the population standard deviation is never given to us Our second condition, which states that the sample size must be less than 30 has also been met because we have a sample size of five. So both conditions have been met to use a t-distribution. So that’s why I chose to use a t-test for this particular example, so like I said before we need to find the critical value, which is going to be 18 value, which separates this area in the tail with the rest of the curve, so in order to find this critical value of T, we need to use a t-table and one thing we need to keep in mind When using our T table is the area in one of the tails. We know we have an area of point zero five in one of the tails. So we need to keep that in mind when using our T table. So if we go and we if we go to our T table and we see that in the top row, we have the area of one tail, so we know that the area of one tail is equal to point zero five, so we know that we have to use this row with the area of point zero five and on the left side of the table. We have our degrees of freedom. The degrees of freedom is always one less than the sample size. Okay, we have a sample size of five, so we know that our degrees of freedom is going to be four, so we’re going to the row with degrees of Freedom 4 and if we look where the rows intersect degrees of freedom for an area of point zero five, they intersect at the T value of two point, one three two, so we have a critical value of two point one three two. So if we go back to our example, let’s take a look at this curve notice how our critical value load lies below the average. It’s below the middle of the curve. So this means that it’s going to be a negative number since it’s at the low end of the curve so instead of a positive two point one three two, we know that our critical critical value is going to be a negative two point 1 3 2 So why is this critical value so important? It’s because it separates this area in the tail with the rest of the curve and this area of the tail, which is in red is called our rejection region and the reason why this rejection region is so important is because in our next step we’re going to perform a test. We’re going to get a T value in our test. And if this T value happens to fall in this rejection region, that means we can reject our null hypothesis, Our null hypothesis says that the average is equal to 100 and if in our test, if our T value lies in the rejection region, we can reject that the average is equal to 100 and we can accept the alternative hypothesis. So this is why this rejection region is so important and this brings us to our next step, which is we need to perform a test step number 4 says to find the test statistic and the test statistic is just going to be a Z value or a T value and like I said before we’re using a t-test or T distribution, so our test statistic is going to be a T value, So if we go back to our example of the formula to find the letter T our test statistic, T is equal to the average of the sample minus. The average of the population all divided by the standard deviation of the sample divided by the square root of the sample size N and notice how the formula for the test statistic T is almost identical if we’re using AZ test as well. So let’s plug everything into our formula. First, we’ll start with X bar. The average of the sample we know the average of the sample can be found just by adding all 5 of these test scores and then dividing by 5 so if we add all of these test scores and divide by 5 we’re going to get an average of 89 so we know at our average for the sample. X Bar is equal to 89 and this is being subtracted by the average of the population, which is the Greek letter. Mu and the average of the population is equal to a hundred so we can plug in 100 for our population average. This is all being divided by the standard deviation of the sample and just to save a little bit of time. I gave this. I gave this to us already. The standard deviation of a sample is equal to fifteen point eight one so we can plug fifteen point eight one in for our standard deviation, and this is being divided by the square root of the sample size n. We know that our sample size is five, and if we plug all of this into our calculator, we’re going to get a value of negative one point five six. This value of negative one point. Five six is our test statistic. So how do we use this value to draw a conclusion to this problem? That’s our last step. We need to draw a conclusion to the problem, so if we go back to our curve, this value of negative one point five six. Where does it lie on this curve? Well, we know that this critical value is negative. Two point one, three two and we know this average, which is directly in the middle as a value of zero. So we know negative. One point. Five six is somewhere between negative two point, one, three, two and zero. So I’m just going to estimate that our test statistic, which is negative One point five six, is somewhere in between the critical value in the middle of the curve. So this is negative, one point five, six so notice how this value of negative one point five six is not in the area of the tail. It’s not in the rejection region, which means that we cannot reject the null hypothesis Our null hypothesis, which says that the average of the average. IQ is equal to 100 We cannot reject that. We fail to the null hypothesis, or or the easy way to put it. Is we have to accept it? We accept our null hypothesis that the average. IQ is equal to a hundred. So what is this question asking? Is there enough evidence to suggest that the average? IQ is lower. No, there is not enough evidence because we accept our null hypothesis that it stays the same. It’s still the average IQ is still equal to a hundred. So I hope this video gave you a better idea on hypothesis testing and there are some other examples that are a little bit different, so that’s why I made some more videos to give you a better understanding. I’m in the top left corner. I have the link for a hypothesis testing example where we use a one tailed test and it also uses proportions and in the top right corner. I have the link for a hypothesis testing video where we use a two-tailed test. I’m also I put the links for my videos on confidence intervals on the bottom of the screen. So check those out If you want. Thank you so much for watching, and I will see you in my next one.