Question

A pediatrician wants to determine the relation that may exist between a? child's height and head circumference. She randomly selects 5 children and measures their height and head circumference. The data are summarized below. Complete parts? (a) through? (f) below.

Height​ (inches), x	25	27.75	26.75	25.5	26.5
Head Circumference​ (inches), y	16.9	17.6	17.3	17.1	17.3

a) Treating height as the explanatory variable, x, use technology to determine the estimates of β0 and β1.

(b) Use technology to compute the standard error of the estimate, se.

(c) A normal probability plot suggests that the residuals are normally distributed. Use technology to determine sb1.

(d) A normal probability plot suggests that the residuals are normally distributed. Test whether a linear relation exists between height and head circumference at a=0.01 level of significance. State the null and alternative hypotheses for this test.

Determine the P-value for this hypothesis test. What is the conclusion that can be drawn?

(e) Use technology to construct a 95% confidence interval about the slope of the true least-squares regression line. What is the lower bound and upper bound?

(f) Suppose a child has a height of 26.5 inches. What would be a good guess for the child's head circumference?

Answer 1

Solution

we will solve it by using excel and the steps are

Enter the Data into excel

Click on Data tab

Click on Data Analysis

Select Regression

Select input Y Range as Range of dependent variable.

Select Input X Range as Range of independent variable

click on labels if your selecting data with labels

click on ok.

So this is the output of Regression in Excel.

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.9888
R Square	0.9777
Adjusted R Square	0.9702
Standard Error	0.0450
Observations	5.0000
ANOVA
	df	SS	MS	F	Significance F
Regression	1.0000	0.2659	0.2659	131.4423	0.0014
Residual	3.0000	0.0061	0.0020
Total	4.0000	0.2720
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	10.9674	0.5475	20.0322	0.0003	9.2250	12.7097
Height	0.2385	0.0208	11.4648	0.0014	0.1723	0.3047

a) Treating height as the explanatory variable, x, use technology to determine the estimates of β0 and β1.

β0 =10.9674

β1=0.2385

(b) Use technology to compute the standard error of the estimate, se.

	Coefficients	Standard Error
Intercept	10.9674	0.5475
Height	0.2385	0.0208

(c) A normal probability plot suggests that the residuals are normally distributed. Use technology to determine sb1

sb1= 0.0208

(d) A normal probability plot suggests that the residuals are normally distributed. Test whether a linear relation exists between height and head circumference at a=0.01 level of significance. State the null and alternative hypotheses for this test.

	Coefficients	Standard Error	t Stat	P-value
Intercept	10.9674	0.5475	20.0322	0.0003

the P-value for this hypothesis test.= 0.0003

Since p-value =0.0003 < 0.01 we reject the null hypothesis and conclude that there is linear relationship between two variables.

(e) Use technology to construct a 95% confidence interval about the slope of the true least-squares regression line. What is the lower bound and upper bound?

0.1723

0.3047

(f) Suppose a child has a height of 26.5 inches. What would be a good guess for the child's head circumference?

head circumference =10.9674+0.2385*height

head circumference = 10.9674+0.2385*26.5

head circumference = 17.3