A classic story involves four carpooling students who missed a test and gave an excuse that they had a flat tire on the makeup test. The instructor asked students to identify which tire went flat. If they really didn't have a flat tire, would they be able to identify the same tire? So the author decided he was going to run a an experiment, and the author asked 41 other students toe identify the tire that they would select, and some of them said left front. Others said right front. Some said Left rear and others said Right rear. And the data that was collected was that 11 people said Left front 15 said Right Front eight said left rear, and six said Right rear. Now that totals up to Onley 40. Even though he asked 41 students, one of the students said That spare tire, So we're gonna leave that out of our data. So this author is making a claim and he believes that the results fit a uniformed distribution. So in order for us to test this claim, we're going to have to construct are null hypothesis and our alternative hypothesis, and you're no hypothesis when you're trying to determine whether observed data fits something that is expected, the statement of fit becomes your null hypothesis. So therefore, our claim is going to be our null hypothesis. So our alternative hypothesis is going to be that the results do not fit a uniformed distribution. Mhm. And the hypothesis test that we're going to run is going to be a chi square goodness of fit test, which requires us to generate a chi square test statistic for our data. And we will have to apply the formula some of observed, minus expected quantity squared, divided by expected. So let's go back up to our data, and the information that we've collected would be classified, as are observed data. Yeah, so now we need to calculate are expected values. And if there were 40 people involved and we would expect a uniformed distribution, that means we would expect each response to get 10 people so we would expect tend to say left front, tend to say right front, tend to say left rear and tend to say right rear. So we're now ready to calculate our chi square test statistic, so we're going to add on to our chart An additional column and we're going to call that column oh minus e quantity squared, divided by e. So we'll take the observed value minus the expected value. We'll get a difference of one. We're going to square that So it's still 1/10, which is that expected value. So we'd get 0.1 and then we'll do 15 minus 10. We get a result of five when we square it. It's 25 divided by the expected value of 10. Yields a value of 2.5. Do the same thing for left rear eight. Minus 10 is negative two. But when we square it, it's positive. Four, divided by the expected value of 10, resulting in 0.4 and then six minus 10 would be negative. Four. When we square that we get positive 16/10 or 1.6 now to calculate the Chi Square test statistic, we will have to add up these values. And when we add up those values, our Chi Square test statistic is 4.6 now. Another component of the hypothesis test is to calculate our P value, and R P value is referring to the probability that Chi Square is greater than the test statistic we just found. Now, to get a better sense of what that's about, we're going to draw a chi square Distribution and chi square distributions are skewed to the right, and their shape is dependent on the degrees of freedom, and our degrees of freedom can be found by doing K minus one. And K represents the number of categories which you've separated your data into. If we go back to our chart, you can see we have separated our data into four different categories, corresponding with the four different tires on the vehicle. So if K is four, then our degrees of freedom will be three. Not only does the degrees of freedom indicate the shape of the graph, but the degrees of freedom also is equivalent to the mean of that chi square distribution. So on our picture we could place the mean, which will be found slightly to the right of the peak on the Chi Square axis. Now, for us to find our P value, we're trying to figure out what's the probability that Chi square is greater than 4.6. So we're trying to determine this shaded region, and in order to do so, the most effective way is to utilize your chi squared cumulative density function in your graphing calculator, and any time you use that function, you have to provide the lower boundary of the shaded area, the upper boundary of the shaded area and the degrees of freedom. So for our data, the lower boundary is the test statistic, the upper boundary. If you imagine that curve continuing infinitely to the right, you're going to get to some extremely high values of Chi Square. So we're going to use 10 to the 99th Power to represent our upper limit, and our degrees of freedom was three. So let me show you where you can find that chi squared cumulative density function. So I'm bringing in my calculator and I'm going to hit the second button and the variables button and select number eight. We're going to put the low boundary of the shaded area, the upper boundary of the shaded area, followed by my degrees of freedom, and I end up with a P value off approximately point 2035 Now there's one more component of a hypothesis test that we could find and that is called your chi square critical value. And to determine that chi square critical value, we're going to look in the back of your textbook. You will find a chi square distribution table and down the left side of the table, you'll find degrees of freedom and across the top of the table, you will find levels of significance, and levels of significance are denoted by the Greek letter Alfa. And we want to run this hypothesis test at a level of significance of 0.5 So we will locate 0.5 across the top of the chart and our degrees of freedom down the side of the chart and where the to correspond or meet up is your chi square critical value, and they meet up at 7.815 So let's recap the three components that we have found so far we have found the Chi Square test statistic to be 4.6. We have found the P value to be point 2035 and we have found our chi square critical value to be 7.815 So what do we do with these values to make a decision about that claim. So when it comes time to make your decision, you can either utilize the P value or you can use the chi square critical value. You do not need to do both. I'm going to show you both and then you can make a determination of which method you prefer. In order to use the P value. You're going to compare your level of significance to your P value. And if your level of significance is greater than your P value than your decision is to reject the null hypothesis. So let's run our test. So our level of significance was 05 and we found RPI value to be 0.2035 So we could say that Alfa is not greater than the P value. So therefore our decision will be fail to reject the no hypothesis. Let me show you how you can use the critical value to arrive at that same decision. To use the critical value, I recommend drawing out another chi square distribution and placing your critical value on that curve. And by placing that on the curve, you have broken your graph into two parts. You have the tail which we're going to define as the reject, the null hypothesis region. And then the other part of the graph is going to be determined or called our fail to reject the null hypothesis region and you're then going to look at your calculated Chi Square test statistic and we found our Chi Square test statistic to be a 4.6. So if 7.8 is right here, then 4.6 would fall back here, which is in the fail to reject region. So again, we end up with the decision that we will fail to reject the null hypothesis. So if we go back to our statements, we are not able to reject this statement. So that's saying it could be true. It might be true. It might not be true, but our evidence don't support throwing it away. So therefore, our conclusion there is insufficient evidence to reject the claim that the results fit a uniforms distribution again. We don't have enough to throw it away, but we don't We're not saying we support it, either. It's just the data is inconclusive and that concludes your hypothesis test