In this problem. We are comparing the Asian population in the Lake Tahoe area to the Asian population in the Manhattan area, and you were given some data about their race and their frequencies. So in the Tahoe region versus the Manhattan region and this was self reported data. And there were 1, 419 Asian, um individuals in the South Lake Tahoe region and off those 131 reported as Asian Indian 118 reported as Chinese 1045 reported as Filipino, 80 were reported as being Japanese, 12 Korean, nine Vietnamese and 24 were other. And then in Manhattan, we had 174 Asian Indian, 557 Chinese, 518 Filipina, 54 Japanese, 29 Korean, 21 Vietnamese and 66 other. And in both cases they add up to a some of 1 419 individuals, and we're trying to conduct a goodness of fit test to determine if the self reported sub groups of Asians in the Manhattan area fit the Tahoe area. So we're going to have to write up a null hypothesis and are null hypothesis is going to read the self reported sub groups of Asians. Yes, in the Manhattan New York area fits that of the Lake Tahoe area. Actually, it was the South Lake Lake Tahoe area, as described in the table. And if we're going to test that hypothesis, I try again. Hypothesis. We need an alternative to fall back on if we reject it. So our alternative hypothesis is going to be the self reported subgroups of Asians in the Manhattan New York area do not fit that of the South Lake Tahoe area as described in the table. And again, we want to run a goodness of fit test. And to start the goodness of fit test, where you're going to have to calculate a Chi Square test statistic. And to calculate that test statistic, you will apply the formula some of observed, minus expected quantity squared, divided by expected. So we're going to go back up and label the Manhattan as what we have observed, and we're hoping it matches the Tahoe region. So we're going to refer to this as being the expected. We're going to need to add on another column, and we're gonna call it observed minus expected a quantity squared all over expected. And the fastest way to get that set of numbers would be to utilize our graphing calculator. So you pull in the graphing calculator and we're going to go to the stat feature and edit, and we're going to clear out any lists better there. And we're going to put the observed values, which are the Manhattan values. Enlist one. And we're going to put the corresponding Tahoe values enlist to. And after you have those in, we're going to sit on top of list three, and we're going to tell it to find the Chi Square test statistic by doing the quantity of the observed values. Enlist one minus the expected values. Enlist to that quantity is squared before it's divided by the expected values enlist to, and we get these values and we're gonna round those to two decimal places as we write them in our table. So our values for the Asian Indian is going to be 14.11 for Chinese will be 1633.23 For the Filipino, it will be 265 77 Japanese 8.45 Korean 24 0.8 Vietnamese 16 and other 73.5. And to find our Chi Square test statistic, we need to total these up or add thes and again, there's a fast way of adding them up. Since they're already in our graphing calculator, we'll bring in our calculator, were going to quit and go back to the home screen hit second step. Scoot over to the math operations and we want to sum up list three and we get approximately 2035.146 So that is our chi Square test statistic 2035 point 15 We now need to find a P value. And when we find our P value, what you're finding is the probability that Chi Square is greater than that test statistic. So in this case, it's greater than 2035.15 and our best bet is to draw a graph to represent the situation which ties all of this together. So we're dealing with a chi square distribution, so it will be a skewed right distribution, and our distribution is going to be determined. The shape of it is determined by the degrees of freedom and degrees of freedom are found by doing K minus one. And the K represents the number of categories that your data has been separated into. So if we go back to our original table, we have broken our data up by race into 1234567 different categories. So therefore, R k value it's seven, and our degrees of freedom will be six. And the degrees of freedom are also telling us what our average is and are mean in this case is six. So on our picture, just to the right of the peak is where you'll find your average, so we could put six on our chi square horizontal axis right around there. Now we want to also plot this test statistic, and that's gonna be way, way, way, way out in the tail. So if I were to keep this going and this curve got closer and closer and closer, we're going to be putting that 2000 35.15 way out here, and we're trying to figure out the probability that were greater than that. Well, there's not much area between the horizontal axis and the curve at that point. But that's where you're gonna find your p value. And when it comes time to calculate that P value, our best bet is to use our chi squared cumulative density function in our calculator. And when you do that, it looks for a lower boundary on upper boundary and the degrees of freedom. And in our picture, the lower boundary of that shaded region is going to be 2035 15 Now the upper boundary. Keep in mind, that does keep on going. So therefore, we're going to pick a very large number. We're gonna say 10 to the 99th power and then our degrees of freedom was six. So we're going to bring in the calculator again, and we're going to quit and clear, and you're gonna hit the second button and the variables button, and it's number eight in the menu, you see, right now. So we're gonna type in our low boundary, followed by a comma or upper boundary, followed by a comma, and then our degrees of freedom, which was six, and the probability or the area to the right of 2035.15 is zero. So our P value is zero. Now that we have our test statistic and we have our P value, we are ready to make our decisions. And our decision is based on what we call our level of significance. And our level of significance usually is 0.5 Every once in a while you'll see a level of significance of 0.1 And based on that level of significance, if the level of significance is greater than your P value, then your decision is going to be to reject the null hypothesis. So if we were testing at the significance level of 0.5 0.5 is certainly greater than zero. So we would reject the null hypothesis if we were running this test at a 0.1 Significance level 0.1 is also greater than zero. So either way, our decision is to reject the null hypothesis. So we're gonna go back to that null hypothesis, and we're going to reject it. So if we're rejecting it, weaken, basically cross it off and say That's not true. So therefore, this is what is true. So we are ready to draw our conclusion. And our conclusion would be that there is sufficient evidence that the self reporting subgroups of Asians in the Manhattan New York area do not fit that of the South Lake Tahoe area, and that concludes your chi square goodness of fit hypothesis test.