Okay. In this question, we have a fighter that shows the distribution of how much married US male adults, just their spouses to manage their finances. Now there is a financial services company that claims that the distribution of how much the married females trust their spouses to manage their finances is the same as the picture. Now we're skeptical as always. So what we do is in order to test the claim, we randomly select 400 married US females and ask each of them how much they trust their spouses to manage their finances. So we have the results of our survey with us. The first column is the response column has always What we do is we create a table to solve these kinds of questions. So we have the response column with us, right? So they completely dressed the first. The first category is completely dressed. What? Yeah, yeah, completely. Trust the next one is trust with certain aspects. Yeah, so let me just write this as they trust them. They trust them. They don't completely trust them. They don't blindly trust them, but they just trust them. The fourth, the third one is that they do not trust them. So this is they don't trust. And in the end, we have the not sure category, not short category. Then we have the observed value. These are the frequencies. Or let me just write these as the observed values. Okay, For the completely trust we have 2 43 2, 43. After that, we have trust with certain aspects. This is 108 then do not trust is 36 and then we have studied. All right now we have a by charged with us, which is the distribution of how much the married male US adults trust and the financial services company claims that the distribution is the same for males as well as females. So the probabilities, the probabilities for these categories will be the same as given to us in the pie chart. So completely trust at 65.6. So let me just write this as 0.656 Trust with certain aspects is 27.8. So this is 0.278 Do not trust is 5.7. So this is 0.57 I'm not sure few people who are not sure this is 0.9% of this is zero point 009 Right? 0.90 point 009 Right? Uh, yeah. So this is your 0.9 Okay, Now, what are going to be our now? An alternative hypothesis. The null hypothesis will be that the distribution mhm. The distribution for us married male adults mean? Mm. Yeah. Adults fits the distribution. Yeah, it's the distribution for us married female adults. All right, what will be the alternative hypothesis? The alternative hypothesis will be that the distribution for us married male adults does not fit the distribution. Four. Yeah. US married female adults. All right, Now what we are going to do over here is the chi square. Goodness of fit test. What is the first time? The first step in this analysis is to find the expected values for all the categories. And what is the formula for that? The formula for that is sample size that is in multiplied by the probability for each category. The probability for each category that is P I. All right, Let us look at this formula in action. This is the column for expected values. This will be the column for expected values. Okay, now, what is a sample taste? It is 400 females, 400 married US females. So what will be the expected value For the first category? It is going to be 400 multiplied by zero point 656 or this is to 62.4 to 62.4. For the next category, it will be 400 multiplied by 0.278 This is going to be 111.2. After that, we have 400 multiplied by 0.57 which is 22.8, 22.8 and then it will be 0.9 multiplied by 400. So this should be three 0.6. These are the expected values. Now. The next step in this analysis is finding the chi square statistic. How do we do that now? There is a formula for this. What are you going to do is in the next column you're going to fill up the values for each category. For each category, you're going to find the difference of observing the expected values. Mhm You square them. You divide this value by the expected value and this particular value this expression we find for all the categories and in the end we sum them up. So this gives us the overall price quest at the stake for our problem. Let us go here. Let us look at this formula in action. So for the first category, what is the difference between observed and expected values? It is too. 62.4 minus 2. 43. This is 19.4. I square this. So this becomes 30. 76.36 and I divide this by the expected value to 62.4. So this is 1.434 1.434 For the next one, we have 111.2, minus 108. The square this and divided by 111.2. So this 0.92 0.92 Then we have 36 minus 22.8. We square this this is 1 74 and divide this by 22.8. This is 7.642 7, 70.642 Then we have the difference between 13 and 3.6. We square this and divide this by the expected value. This is 24.54 24.54 Again, we have solved enough questions to see that these the addition of this anti column is very high. It goes into 30 or 32 or something right 32 or 33 something so we can, without any analysis, say that the distributions are different. We will reject Donald Hypothesis, but still let's verify this by doing the proper analysis by using the proper method I some all of these up. So this is 24.54 plus 7.642 plus 0.92 plus 1.434 This is 33.71 So my overall chi square statistic is 33.71 Now, if I want to reject my null hypothesis or not, how do I decide this? I decided this with by the help of two methods. Okay, there are two methods to do this. The first one is the P value method And the second one is the critical funding method. Now, in order to do both of these methods, I need the degrees of freedom and the degrees of freedom is given by the formula. Yeah, number of categories. Number of categories. Yeah, minus one. All right. How many categories do I have? If I look at this, there are just four categories here. So this is going to be four minus one or this will be three. So now I have my chi square value. I have my degrees of freedom. Now, the first matter that I'm using is the P value matter. What is the alpha for this question? The alpha is 0.1. The alpha is 0.1. Now I will find the p value. And if my P values less than Alpha, I will reject final hypothesis or else I will fail to reject it. Now, in order to find the p value, I can use either the Chi square table. But the thing over here is that if I use the chi square table, I won't get the exact value. I will get an approximate range which will still be good enough to reach an answer. But if you use a statistical software like SPS s or R or python or Excel, you will get the exact value. So this is what I'm doing. I'm using an online tool, a way that I have. My chi square statistic is 33 point. This is 33.71 and degrees of freedom is so this is 33.71 and my degrees of freedom is three. And my alpha is 0.1. I hit calculate, and the P value that I get is much less than 0.1 So let me just write this zero from a P value is approximately 4 to 0. My p value is approximately equal. Yeah, my P value is approximately equal to zero now, Since my P value is less than alpha, this will suggest that I will reject. I will reject my final hypothesis H north. Right now this was a P value method. How do I find this? Using the critical value method, I need to find the critical chi Square statistic, the critical value. So what will be my critical value? My degrees of freedom is three and my alpha is 0.1. So I just put in these values My advice. One my degrees of freedom is three and I hit calculate and I get 6.251 My critical value turns out to be 6.251 So what exactly does this mean? If this is my graph for the class quested to stick, let's say that this is my critical value 6.25 and 6.251 to the right of this, I have the rejection region. This is going to be the rejection region. Okay, so if I get a chi square statistic that that is to the right of 6.251 that is, it is greater than 6.251 It will fall here in the rejection region and I will reject minor hypothesis. I can see that my value is 33.71 which falls way towards the right, which is much bigger than 6.251 Hence, I will reject minor hypothesis. See the same result that we got from the P value method. Now, if we go up and look at the non hypothesis, the hypothesis was that the distribution for us married male adults fix the distribution for us married female adults. We are rejecting this. So how do I frame my answer? In the end, my answer will be that ad 10% significance level at 10% significance level. I have enough statistical evidence. Statistical? Yeah. Evidence to suggest that the distribution yes, of men married us adults, US adults, mhm of male married US adults. Yeah, and the distribution of female married us Adults are different, are different. Yeah, are different. Which means if you look at the question, what was the claim? The claim was the financial services communicate the distribution of how much US married women illustrates their spouses. The man is the finance is the same. The claim of the financial services company. What we are effectively saying is that the claim? Yeah. Of the financial services company of the financial services company. Mhm. It doesn't seem to be right. Yeah. Doesn't seen yeah to be right. Okay, so let me just go with this what we had first of all, we had a nail in the alternative hypothesis. The null hypothesis was that the distribution for US male adults about how much they trust their female counterparts to manage their finances is the same for the US married females, farmers that trust their male counterparts. And the alternative hypothesis was that the distribution for us married males in the US married females was different. We had a sample of 400 a females. We had the probabilities, the pie chart was given to us, and based on the probabilities and the sample size, we calculated the expected values. The formula was sample size multiplied by the probability for each category. This gave us the expected values for this category, and in the end, we wanted to find the guy Specialist IX. So for all the categories we applied this formula, will you Oh, my inner city hall square upon e, we sum them all up, and we found that our chi square statistic is 33.71 Now, we could have just said over here that yes, this, uh, we have project and a hypothesis, since this guy's for value is very high. But we use this two methods to find this to verify our answer. The first one was a p value matter. We found that with the degrees of freedom three, our P value was very close to zero. Hence, we reject around a hypothesis and even using the critical value matter for Alpha of 0.1, we found that the critical value 6.25 and our value of 33 falls in the rejection reason. So, yes, Beyonce's we're matching now. The main thing was to frame the correct answer. In the end, the conclusion. The conclusion was that at 10% significance level, we had enough statistical evidence to suggest that the distribution of male married US adults and the distribution of the female married US adults were different. Which means that the claim of the financial services company doesn't seem to be right there. Claim is not correct. And this is how we go about doing this question, Yeah.