For this problem. We're given data for waist size and percent body fat, and we want to see if there is an association between these two variables. And so for part, A were asked to test inappropriate hypothesis and state our conclusion. But first, let's take a look at the data and see if the the conditions necessary for inference are satisfied. So I've used Excel, and I've entered the data for a waist size and body fat here. And so if I go to data analysis, select regression and then for the Y variable, it's like body fat. And then for the X variable, I select waist size and then I'm going to check 95% confidence level. And I'm also going to check residual plots, line fit plots and normal probability plots. So then we hit okay, and let's look at the normal probability plot first. So this is the normal probability product plot for the residuals, and we're looking to see if it forms a fairly straight line. It's pretty good, a little wobbly, but it's pretty. Much of that resembles a straight line, which indicates that the data is nor the residuals air normally distributed the errors are normally distributed, and usually the first thing we look at is this. Just the scatter plot of the raw data to see if it actually looks linear. So if we just look at the blue data, that's the That's the data given in the question. The oranges is a predicted line. If you look at the blue, it does look like pretty linear data like it's kind of painting the picture of a line there. And then if we look at the residuals plot of body, it's it's the error versus the waist size and what we're looking for. Here's randomness. We don't want to see any patterns or clumping or any trans in the data. This looks pretty nice and random, and we don't really see any fanning out or fanning in. So that suggests that we've also met the the condition for for independent data as well as constant variance or or equal variance. So feeling that we've satisfied the conditions, we can go ahead and look at the regression analysis output and actually, why don't we make our hypothesis test now? Our hypotheses now? So the no hypothesis is that the slope of the relationship is zero, which means that there is no association between waist size and percent body fat, and the alternative hypothesis is that the slope is not equal to zero, which means that there would be an association between with size in percent, body fed and so moving back to the regression analysis output. We have a non zero coefficient for the slope estimated and a T value of 7.4 and a P value of about one out of a 1,000,000 so very close to zero, much less than 1% or something like that. So based on that small P valley, we would reject the no hypothesis and conclude that there is enough evidence to suggest an association between waist size and body percentage fat and next for Part B, were asked to give a 95% confidence interval for the mean percent body set for people who have 40 inch waist. So such an interval is of this form. So let's tackle these three parts one by one, absolutely predicted value for Why sequel to But why intercept plus the slope times x seven you and so this is our predicted value for percent body fat and accept New is the 40 inch waist. So using our regression analysis out But we have the two coefficients We have the Y intercept of minus 60.4165 and the slope of 2.167 So negative 60 point for two plus 2.17 times 40. And so we have predicted average percent Body fat sequel to about 26.38 Okay, so we have this value. Is this valley here? No, for the critical value. So if if you look at the question, it's it's basically 20 samples, so 20 people are are used in the sample, so the degrees of freedom is 18 and we're doing a 95% confidence interval. So Alfa is 0.5 So if you look this up on an excel weaken dio equals t dot I envy dot to t enter 0.5 for the probability and 18 for the degrees of freedom. And we get 2.10 And now for the standard error on the predicted why value and that is of this form. So from the output and excel, the standard air on the slope is zero point 292 and the X seven new is 40. That's the 40 inch waist, and we need to find the average of the waist sizes from the from the sample. And if you calculate that average from the data given in the question to get 37 point 05 and now for the residual standard deviation, that's in the output. So that is, uh, this value right here, 4.6 to 4. So that's 4.624 squared, then divided by 20 which is the sample size square root of that, and this comes out to 1.346 So now we have everything we need to make our interval. We have this value just value. We have the critical value, and we have a standard error on the Y estimate. So the interval is 26.38 plus or minus 2.1 times 1.346 which comes out to 23.55 2 29 0.1 three. So what this means is we're 95% confident that present that the average percent body fat for people who have 40 inch waists, is between 23.55% and 29.13%.