Question

Prehistoric pottery vessels are usually found as sherds (broken pieces) and are carefully reconstructed if enough sherds can be found. Information taken from Mimbres Mogollon Archaeology by A. I. Woosley and A. J. McIntyre (University of New Mexico Press) provides data relating x = body diameter in centimeters and y = height in centimeters of prehistoric vessels reconstructed from sherds found at a prehistoric site. The following Minitab printout provides an analysis of the data. Please show steps for part C

Predictor	Coef	SE Coef	T	P
Constant	-0.236	2.429	-0.09	0.929
Diameter	0.7689	0.1281	5.33	0.001
S = 4.11870	R-Sq = 77.0%

(a) The standard error S_e of the linear regression model is given in the printout as "S." What is the value of S_e?
4.11870

(b) The standard error of the coefficient of the predictor variable is found under "SE Coef." Recall that the standard error for b is S_e/√Σx² – (1/n)(Σx)². From the Minitab display, what is the value of the standard error for the slope b?

.1281


(c) The formula for the margin of error E for a c% confidence interval for the slope β can be written as E = t_c(SE Coef). The Minitab display is based on n = 9 data pairs. Find the critical value t_c for a 90% confidence interval in the relevant table. Then find a 90% confidence interval for the population slope β. (Use 3 decimal places.)

tc
lower limit
upper limit

Answer 1

Solution:

Given:

Predictor	Coef	SE Coef	T	P
Constant	-0.236	2.429	-0.09	0.929
Diameter	0.7689	0.1281	5.33	0.001
S = 4.11870	R-Sq = 77.0%

Part a) The standard error S_e of the linear regression model is given in the printout as "S." What is the value of S_e?
S_e = 4.11870

Part b) what is the value of the standard error for the slope b?

SE_b = 0.1281

Part c) Find the critical value t_c for a 90% confidence interval in the relevant table. Then find a 90% confidence interval for the population slope β.

n = 9

thus df = n - 2 =   9 - 2 = 7

two tail area = 1 - c = 1 - 0.90 = 0.10

t_c = 1.895

Margin of Error is:

E = t_c(SE Coef).

E = t_c * (SE_b).

E = 1.895 * 0.1281

E = 0.2427495

E = 0.24275

Thus, a 90% confidence interval for the population slope β is given by:

( b - E , b + E )

From above table , we have b = 0.7689

thus

Lower Limit = b - E

Lower Limit = 0.7689 - 0.24275

Lower Limit = 0.52615

Lower Limit = 0.526

and

Upper Limit = b + E

Upper Limit = 0.7689 + 0.24275

Upper Limit = 1.01165

Upper Limit = 1.012