Hypothesis Testing - Statistics

Welcome to my video on Hypothesis testing. Here in this example it says the average IQ for the adult population is 100 with a standard deviation of 15. A researcher believes this value has changed. the researcher decides to test the IQ of 75 random adults. The average IQ of the sample is 105. Is there enough evidence to suggest the average IQ has changed? So there are five steps in order to perform an hypothesis test. So these five steps can be used for any type of hypothesis testing example. So let me show you what these five steps are. The first step says to state the null and alternative hypothesis. The null hypothesis is written with "H sub zero" or "H naught" and the alternative hypothesis is written with "H sub 1" or sometimes it's written with "H sub A". So if we go back to our example let's start with our null hypothesis. We need to state our null hypothesis. Now the null hypothesis is what is currently believed to be true. So in this particular example it says the average IQ the adult population is 100. So it is currently believed that the average IQ is 100. So that is our null hypothesis. The average for the population which is is written with the Greek letter Mu is equal to 100. And the null hypothesis is always written with an equal sign so if you don't have an equal sign in your null hypothesis then you have done something wrong. So now let's state our alternative hypothesis. The alternative hypothesis "H sub 1" or "H sub A" it doesn't matter. Is always what is being claimed. What is being claimed? In this particular example it says a researcher believes He is claiming that this value has changed. Ok and these two words are really important "has changed" He doesn't know if the average has gone up or if the average has gone down. He doesn't know that but he just believes it has changed He believes that the average is not 100 anymore. So the alternative hypothesis what is being claimed is that the average IQ is not 100 so it does not equal 100. And the alternative hypothesis is always written with a not equal sign or greater than or less than sign. So anytime you see the alternative hypothesis written with a not equal sign. This mean we are going to perform a two tailed test. A two tailed test. And I'm going to explain what this means later on in this video. But anytime you see a not equal sign in your alternative hypothesis that means you're going to perform a two tailed test. So now that we have stated our null and alternative hypothesis let's move on step number 2. So if we go back to our steps it says step number 2 is choose our level of significance which is written with the Greek letter Alpha. So we need to choose our level of significance for this example. Now sometimes the level of significance is given to you in the problem but if not then you need to choose the level of significance. So what does this mean? The level of significance is just the area in the tails. So let me draw a picture for you so I'll show you what this means. So here I drew for you the normal curve and the level of significance is just the area in the two tails. Remember this is a two tailed test so we have to use both of the tails. So the level of significance is just the area in both of the tails. And I shaded the area in red. And the reason why I chose a level of significance of .05 is because .05 seems to be the most common level of significance used and it's not given to us in this example. So I just chose the most common level of significance which is .05. So these two tails are symmetrical so that means they have the same exact area. So if both of them combined have an area of .05 that means the area in one of the tails has to be .025 and the area in the other tail also has to be .025. And if you add these two areas together we get our level of significance of .05. One more thing to point out. We know the total area under the curve is equal to 1 or 100%. So if the area of the tails is equal to .05 that means the middle of the curve (the area) has to be equal to 95% or .95. So now that we've stated our level of significance let's move on to step number 3. Step number 3 says to find the critical values. Find our critical values. Now the critical values can be a z-value or a t-value so let me go to our example and I'll explain when to use "z" and when to use "t". We always use a z-value if the population standard deviation is known. So in this example it says the average IQ for the adult population is 100 with a standard deviation of 15. So the population standard deviation is given to us. The population standard deviation which is written with Sigma is equal to 15. So since this is given to us we are going to use a z-test instead of a t-test. So we need to find the critical values using "z". And the critical values are just the z-values that separate the area shaded in red or the area of the tail and the middle area. So these critical values are just area that separates the tails and the middle of the curve. So in order to find these critical values we are going to use a "z" table. So there is lots of information we can use to find the values in the z table. This is a 95% interval so we can use a 95% confidence interval to find the z-values. Or we could just use the area in one of the tails which is .025. We can also use that area to find the z-value. So let's just go to our z-table and notice in our z-table if we have a confidence level of 95% that means the area in one of the tails is going to be .025. And that will give is a z-value of 1.96. So our critical value of a 95% confidence level or an area of .025 in one of the tails. The critical value is going to be 1.96 because that's that our z-value is. So if we go back to our example we know that this critical value is going to be equal to 1.96 and the critical value on the low end of the curve is below the average so it's going to be -1.96. So why are these critical values so important? It's because they separate the area in red from the middle of the curve. And this area in red is what we call the "rejection region". And I wrote this down for you as well. The area that is red or the area of the tails is called our "rejection region". And the reason why this "rejection region" is so important is because we're going to perform a test in our next step. And that test is going to give is a z-value for our sample. And that z-value for our sample if it falls in any of the "rejection regions" or if it falls in any of these areas in red then that means we can reject our null hypothesis. OK if our test that we perform if the z-value falls in the "rejection region" that means we reject our null hypothesis which means our null hypothesis which says the average is equal to 100. We reject that. That's why this is so important because it allows to draw a conclusion after we perform our test. So that brings us to our next step. Step number 4. So once again here are all of our steps and step number 4 says to find the test statistic. So what this means is we're going to find a z-value for the given sample. Now remember we chose to use a z-value because the population standard deviation was known. Sometimes you need to use a t-test so I'm going to go over another example when we need to use a t-test so stay tuned for that. But for this particular example we need to use a z-test so let's go back and let's find our value of z. So the formula to get our "z" test statistic is equal to the average of the sample which is written with x bar minus the average of the population which is written with the Greek letter Mu all divided by the standard deviation of the population divided by the square root of the sample size n. So let's go ahead and plug everything into our formula. Our value of x bar is our sample average and if we go to our original problem it says the average of the sample is 105 so we know the average of the sample is equal to 105 so we can plug in for x bar 105. And this is being subtracted by the population average the Greek letter MU and we know that the Greek letter Mu what we believe to be the average is equal to 100. This is the population average so we're going to plug in 100 for he Greek letter Mu. And our population standard deviation is also given to us so if we go back to our original problem it says that the average for the adult population is 100 with a standard deviation of 15 so the population standard deviation is equal to 15. So we have a 15 for the population standard deviation all divided by the square root of the sample size n and I think we had a sample size of 75 (yup!) The researcher decided to test the IQ of 75 random adults. So that means we have a sample size of 75. So we'll plug in 75 for our sample size n. So I'm just going to plug all of this into our calculator and we get a value of 2.89 for "z". So this value of 2.89 is our test statistic. So how can we use this test statistic of 2.89 and draw our conclusion? And that brings us to our last step. We need to draw a conclusion using the test statistic. So we know this value of 2.89 is definitely in the red area of the curve. Because this critical value of 1.96 is lower than 2.89. So 2.89 is certainly above 1.96. I'll just draw a line just to estimate where it is. Somewhere around here this is around 2.89 which is certainly in the red area or in the "rejection region". So because our value lies in the "rejection region" we CAN reject the null hypothesis. OK so let me write this for you in green. Based on our test we CAN reject our null hypothesis which means we reject that the average is equal to 100 and we can accept the alternative hypothesis. We can accept the alternative hypothesis. We can accept that the average is not equal to 100. We're not saying that it's true but we can accept it. So in this problem it says "Is there enough evidence to suggest the average IQ has changed"? There is enough evidence because we can accept the alternative hypothesis that the average has changed. It's not equal to 100.

Channel: Math Meeting

Views: 711,591

Keywords: Statistics (Field Of Study), Statistical Hypothesis Testing, Hypothesis testing, Mathematics (Field Of Study), Math Meeting

Id: 0XXT3bIY_pw

Length: 13min 32sec (812 seconds)

Published: Tue Apr 07 2015