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.

Predictor	Coef	SE Coef	T	P
Constant	-0.197	2.429	-0.09	0.929
Diameter	0.8175	0.1227	5.33	0.013
S = 4.19430	R-Sq = 78.4%

(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?


(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?


(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 = 14 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	-0.197	2.429	-0.09	0.929
Diameter	0.8175	0.1227	5.33	0.013
S = 4.19430	R-Sq = 78.4%

Part a) What is the value of S_e?

S_e = S = 4.19430

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

The standard error for the slope b = SE Coef_Diameter = 0.1227

Part c) find a 90% confidence interval for the population slope β.

t_c is t critical value for c = 90%  confidence level

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

df = n - 1 =  14 - 1 = 12

Look in  t table for df = 12 and two tail area = 0.10 and find t critical value

t_c = 1.782

Formula for confidence interval for slope β is:

where

b = slope coefficient = 0.8175

Thus

and

Thus we get: