1. Mandel looks at the relationship between the amount of...

1. Mandel looks at the relationship between the amount of nickel (the regressor) and the volume percent austenite in various steels. data on excel file HW6 Austenite. a. Use MINITAB to do a scatter plot. Does it seem there is a relationship between x and y? b. Estimate the intercept and the slope write the regression line c. What is the value of the coefficient of determination $R^2$? Interpret. d. Use a t-test to test the significance of the slope coefficient at $\alpha = 0.05$. Comment on your results e. Construct the ANOVA table and test for significance of regression. Comment on your results and relationship to your results on (d) f. Construct a 95% CI for $\beta_1$. Comment on the relationship of these CI and your findings in parts (d) and (e). g. Calculate and plot the 95% Confidence and prediction bands.

Transcript

00:01 Once again welcome to a new problem.

00:04 This time we're dealing with regression.

00:08 We're dealing with regression and when it comes to the equation we have the regression equation which is b0 plus b1 x.

00:19 B not is the intercept and b1 is the slope.

00:28 Remember x is the independent.

00:35 Variable x happens to be the independent variable and y happens to be the dependent variable and when you look at the dependent variable this is the response variable and on top of the response variable the independent variable is the explanatory variable the relationship between the explanatory variable x and the response variable y is governed by the correlation coefficient which for the most part runs from negative 1 up until 1 with 0 showing no relationship and 1 negative 1 showing a perfect negative relationship and 1, shows a perfect positive relationship.

01:53 So you have perfect negative relationship, perfect positive relationship.

01:58 Anything above 0 .7 or below negative 0 .7 is considered a strong relationship and then in between we have moderate relationships.

02:13 We're looking at a new problem and our goal is to analyze the relationship between hydrocarbons so there's a relationship between hydrocarbons and purity of oxygen so you're looking at hydrocarbons and of course you're also looking at the purity of oxygen this is a chemical distillation process and we have 20 observations we want to look at the equation outcomes like we say that as a percentage x is the hydrocarbon level x is the hydrocarbon level and y happens to be the purity these are the numbers we have 20 observations the first step in this problem is if we want to determine the direction of the correlation coefficients.

03:32 So we're going to say as the level of hydrocarbons, as the level of hydrocarbon increases, so does the level of purity.

04:00 Again, we want to see this is purity of oxygen.

04:06 The purity of oxygen as the levels of hydrocarbons go up so the level of purity of oxygen also goes up so in a nutshell this is a positive relationship so you're looking at a typical positive relationship input b how the question is we want to determine the r value and also test for the significance.

04:44 So the r value is the same as 0 .9369.

04:58 The r value is the same as 0 .93669.

05:05 And we want a test for the significance of the r value.

05:11 So it means that we need a test statistic which is a t test.

05:17 First of all, we're assuming with the now hypothesis, r is zero and the alternative hypothesis r is not equal to zero.

05:27 So in the initial part, no significance.

05:33 And then in the second part, they're saying there is significance.

05:38 So there are two options here.

05:40 There's an option of no significance and then there is also significance.

05:44 So r radical 1 minus n all of 1 minus r squared.

05:53 So this is the formula that we're going to use.

05:59 We have to do the math for these numbers.

06:04 The formula for the significance, again, it's a hypothesis test and we've identified in our hypotheses in terms of significance.

06:27 So in terms of the significance level that's what's happening in the problem is the correlation coefficient we're handling.

06:38 So we're just going to go ahead and plug in the relevant numbers.

06:42 The first one is the r value which is telling us that we have 0 .9369, 0.

06:53 0 .9 to be 669 and of course we have we have a sample size of 20 we have a sample size of 20 so that means we're gonna well actually this one this one should be n minus 2 if i'm not mistaken this one is the correlation coefficient we're testing so we're going to have value.

08:25 So this should be instead of using r, we're going to use the population correlation coefficient.

08:34 That's what we're going to use.

08:36 So instead of using r, we're going to be using r.

08:39 So this is r.

08:41 And that's the number we're using.

08:43 And we have the test statistic that's relevant for this particular problem that's that we're dealing with.

08:56 So then again we have 20 data points.

09:01 Just remember that we have 20 data points.

09:04 So radical, this is going to be radical n minus 2.

09:08 That's the formula for our test statistic.

09:12 N happens to be 20.

09:14 So 20 minus 2 divided by 1 minus r squared.

09:19 R squared is 0 .8.

09:22 1.

09:25 Actually 0 .874.

09:29 That's what we're saying.

09:31 R squared is .8774.

09:35 This allows us to do the computation.

09:41 So then the value is going to be 0 .9366 radical 18.

09:55 We want to divide that by 1 minus.

10:00 9366 .0 .0.

10:07 So there's another radical right there.

10:12 So this is point, this is also a radical.

10:24 So radical point 0.

10:29 So we get radical 0634.

10:35 And then we'll go ahead and do the actual math for this problem so let's see how the numbers play out...

Question

Please give Ace some feedback