Question

The following data is representative of that reported in an article with x = burner-area liberation rate (MBtu/hr-ft2) and y = NOx emission rate (ppm):

x 100 125 125 150 150 200 200 250 250 300 300 350 400 400
y 150 150 170 220 190 330 270 390 420 450 400 590 610 680
(a) Does the simple linear regression model specify a useful relationship between the two rates? Use the appropriate test procedure to obtain information about the P-value, and then reach a conclusion at significance level 0.01.
State the appropriate null and alternative hypotheses.
H0: β1 = 0
Ha: β1 ≠ 0H0: β1 = 0
Ha: β1 > 0 H0: β1 ≠ 0
Ha: β1 = 0H0: β1 = 0
Ha: β1 < 0
Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to three decimal places.)
t =
P-value =
State the conclusion in the problem context.
Reject H0. There is no evidence that the model is useful.Fail to reject H0. There is evidence that the model is useful. Reject H0. There is evidence that the model is useful.Fail to reject H0. There is no evidence that the model is useful.
(b) Compute a 95% CI for the expected change in emission rate associated with a 10 MBtu/hr-ft2 increase in liberation rate. (Round your answers to two decimal places.)
, ppm

Answer 1

The hypothesis test for the relationship between burner-area liberation rate and NOx emission rate results in rejecting H0 at the 0.01 significance level, indicating a useful model. The p-value of 0.026 suggests a linear relationship is present. Moreover, the 95% confidence interval for the expected change in emission rates would require additional statistics for calculation.

Step-by-step explanation:

To determine if there is a useful relationship between the burner-area liberation rate (x) and NOx emission rate (y), we conduct a hypothesis test for the slope of the linear regression. The appropriate null (H0) and alternative (Ha) hypotheses to test for a nonzero relationship are:

• H0: β1 = 0 (There is no linear relationship)
• Ha: β1 ≠ 0 (There is a linear relationship)

Given the calculated p-value of 0.026, which is less than the significance level of α = 0.01, we would reject the null hypothesis. Thus, we conclude that there is evidence that the model is useful.

The specific conclusion in the context of this problem is: Reject H0. There is evidence that the model is useful.

For part (b), to compute the 95% confidence interval for the expected change in emission rate associated with a 10 MBtu/hr-ft2 increase in liberation rate:

Let's denote the estimated change in y for a 1-unit change in x (burner-area liberation rate) as β2. The 95% CI for β2, associated with a 10-unit change in x, can be calculated as β2 ± (t* · SE), where t* is the t statistic for a 95% confidence level and SE is the standard error of the slope. This calculation would require further statistics beyond the given p-value, specifically the standard error and the relevant t statistic from a t-distribution based on the degrees of freedom (n - 2, where n is the number of data points).

Answer 2

Using a significance level of 0.01 and a p-value of 0.026, there is sufficient evidence to reject the null hypothesis and conclude that there is a significant linear relationship between the burner-area liberation rate and the NOx emission rate.

Step-by-step explanation:

To determine whether the simple linear regression model specifies a useful relationship between the burner-area liberation rate (x) and NOx emission rate (y), we test the null hypothesis H0: β1 = 0, which states that there is no relationship between x and y (i.e., the slope is zero), against the alternative hypothesis Ha: β1 ≠ 0, which suggests that there is a relationship (i.e., the slope is not zero).

With the given p-value of 0.026, which is less than the significance level of 0.01, we reject the null hypothesis (H0). This indicates that there is evidence to suggest a significant linear relationship between the burner-area liberation rate and the NOx emission rate.

For part (b), we would typically use the estimated regression coefficients and their standard errors to compute the 95% confidence interval for the expected change in emission rate associated with a 10 MBtu/hr-ft² increase in liberation rate. However, since the necessary regression output is not provided, we cannot calculate these values explicitly.