16. The data in the table is the computer output from a...

16. The data in the table is the computer output from a least-squares linear regression analysis on a set of 20 sample for y = a + b x Predictor Coef SE Coef Constant 2.5 0.134 x 0.164 0.057 Student A wants to test H0: ? = 0 vs. Ha: ? < 0. At the ? = 0.01 level of significance, is there sufficient evidence to conclude a linear relationship with slope < 0 between y and x? Use t-test and critical region method. 17. For the data in the table: 1) Find the linear regression line, ? = ? + ?x 2) Find the estimate of the standard deviation of error, s? 3) What is the standard error of the slope? 4) What is the t-statistic for testing the null hypothesis that the population slope in this setting is 0? 5) Test H0: ? = 0 vs. Ha: ? ? 0 At the ? = 0.05 level of significance, is there sufficient evidence to conclude a linear relationship between y and x? Why? Use t-test and P-value method. y x 29 60 32 28 13 32 17 30 4 0 18. A local newspaper surveyed 100 customers, using simple random sampling, for the service they received from a large electric utility. Among the sampled customers, 73 percent say they are very satisfied. Based on these findings, can you reject the local newspaper's hypothesis that at most 80% of the customers are very satisfied? Use a 0.05 level of significance, z-test and P-value method.

Transcript

00:01 So i have some questions here.

00:01 And the first one is, we're given that there's a sample of size n equals 20.

00:07 To make a model, y hat equals a plus b x, where, and then we had, there was a model created, and then we were able to get the coefficients of the model, and they're standard errors.

00:20 The constant was 2 .5, and then the slope with coefficient 0 .164.

00:27 So it's x, that's our model.

00:29 And we want to test the null hypothesis that the slope coefficient b, that's given as beta, is equal to 0 versus the alternative that beta is greater than 0.

00:55 Because here we see it's positive, right? so it's greater than 0.

00:59 The question is, is that significantly greater than 0? that's what we're looking for.

01:03 And it wouldn't make sense to be less than 0, because it's not even negative, right? so it doesn't work.

01:10 And we're going to test this set of hypotheses at the alpha of 0 .01 level of significance.

01:16 And so we're going to use the critical region approach.

01:18 So we need to get our critical value.

01:19 The critical value associated with this is 2 .552.

01:24 And i got that by using the following spreadsheet function, tinv.

01:29 And then you put in the two -tailed alpha that you'd split.

01:32 So this function works that, it puts in, you're assuming your alpha, it's assuming your alpha is two -tailed, meaning you're cutting them into both tails.

01:40 But we just want one tail, just strictly greater than 0.

01:44 So 0 .01, and 0 .01 here.

01:48 So this function assumes two tails.

01:50 You have to make 0 .02.

01:51 And then you put in your degrees of freedom, which is given as n minus 1, n minus 2, excuse me, for regression.

01:58 And so that's 18, 18 degrees of freedom.

02:02 And that gives us the value.

02:04 And then what we're going to do is we're going to reject our null hypothesis, h0, if our test statistic is greater than this critical value, than this t critical value.

02:20 And generally, you absolute value your test statistic, because sometimes it can be negative.

02:25 But here it's positive, so it doesn't really matter.

02:27 So the way you get your test statistic is you take your coefficient, divide it it's the standard error of said coefficient.

02:35 So it's 0 .164 divided by 0 .057.

02:40 We just do that.

02:41 And that gives us our test statistic.

02:44 And it ends up being 2 .877, which is greater than our t value here, our critical value.

02:52 Therefore, we do reject.

02:55 So reject my hypothesis and say, we do have evidence to say that the coefficient is significantly greater than 0.

03:01 All right.

03:02 The next question is that we're given some data.

03:09 They kind of try and pull one on us.

03:11 They put y and x.

03:12 But what we want to do is make a linear regression line, y hat equals a plus bx.

03:19 So our x is still the independent, y is still the dependent variable.

03:23 And we're going to estimate the standard error.

03:28 So we're going to do a few things.

03:30 First thing we do is get the standard error of the estimate, s sub e, which is standard deviation of the error.

03:38 We're going to get that.

03:39 We want to get the standard error of the slope.

03:42 So s, s, e, beta.

03:49 And then we want to make the test statistic for testing the hypothesis that the population slope is 0.

03:57 So in other words, what we're doing is we're conducting another hypothesis test, where this time the null is that the slope is equal to 0 versus the alternative, the beta is not equal to 0.

04:09 So while this was a one -tailed test, strictly greater than, this is going to be a two -tailed test.

04:16 And then we're going to test this hypothesis at the alpha 0 .05 level of significance.

04:21 And we're going to use the p -value approach, which means we're going to get the p -value, and we're going to reject our null hypothesis if our p -value is less than 0 .05.

04:40 The t -statistic, we're going to get that doing the same thing.

04:44 We're going to take the slope coefficient divided by the standard error.

04:48 And let's go ahead and do this...

Question

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