Question

height and head circumference. The data are summarized below. Complete parts​ (a) through​ (f) below.

Height​ (inches), x	2727	27.527.5	26.526.5	2626	24.524.5
Head Circumference​ (inches), y	17.517.5	17.517.5	17.317.3	17.317.3	17.117.1

​(a) Treating height as the explanatory​ variable, x, use technology to determine the estimates of

beta 0β0

and

beta 1β1.

beta 0β0almost equals≈b 0b0equals=

nothing ​(Round to four decimal places as​ needed.)beta 1β1almost equals≈b 1b1equals=

nothing ​(Round to four decimal places as​ needed.)

B Use technology to compute the standard error of the​ estimate,

s Subscript ese.

s Subscript eseequals=

​(Round to four decimal places as​ needed.)

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

s Subscript b 1sb1.

s Subscript b 1sb1equals=  

​(Round to four decimal places as​ needed.)

​(d) A normal probability plot suggests that the residuals are normally distributed. Test whether a linear relation exists between height and head circumference at the

alphaαequals=0.010.01

level of significance. State the null and alternative hypotheses for this test.

Choose the correct answer below.

A.

Upper H 0H0​:

beta 1β1equals=0

Upper H 1H1​:

beta 1β1not equals≠0

Your answer is correct.

B.

Upper H 0H0​:

beta 1β1equals=0

Upper H 1H1​:

beta 1β1greater than>0

C.

Upper H 0H0​:

beta 0β0equals=0

Upper H 1H1​:

beta 0β0greater than>0

D.

Upper H 0H0​:

beta 0β0equals=0

Upper H 1H1​:

beta 0β0not equals≠0

Determine the​ P-value for this hypothesis test.

​P-valueequals=

​(Round to three decimal places as​ needed.)

What is the conclusion that can be​ drawn?

A.

RejectReject

Upper H 0H0

and conclude that a linear relation

existsexists

between a​ child's height and head circumference at the level of significance

alphaαequals=0.010.01.

Your answer is correct.

B.

Do not rejectDo not reject

Upper H 0H0

and conclude that a linear relation

does not existdoes not exist

between a​ child's height and head circumference at the level of significance

alphaαequals=0.010.01.

C.

RejectReject

Upper H 0H0

and conclude that a linear relation

does not existdoes not exist

between a​ child's height and head circumference at the level of significance

alphaαequals=0.010.01.

D.

Do not rejectDo not reject

Upper H 0H0

and conclude that a linear relation

existsexists

between a​ child's height and head circumference at the level of significance

alphaαequals=0.010.01.

​(e) Use technology to

construct

a​ 95% confidence interval about the slope of the true​ least-squares regression line.

Lower​ bound:

Upper​ bound: 0.351

​(Round to three places as​ needed.)

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

A good stimate of the​ child's head circumference would be inches.

​(Round to two decimal places as​ needed.)

Answer 1

I used R software to solve this question.

R codes:

> x=c(27,27.5,26.5,26,24.5)
> y=c(17.5,17.5,17.3,17.3,17.1)
> fit=lm(y~x)
> summary(fit)

Call:
lm(formula = y ~ x)

Residuals:
1 2 3 4 5
0.062264 -0.007547 -0.067925 0.001887 0.011321

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 13.66792 0.61494 22.227 0.000199 ***
x 0.13962 0.02336 5.976 0.009378 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.05379 on 3 degrees of freedom
Multiple R-squared: 0.9225, Adjusted R-squared: 0.8967
F-statistic: 35.71 on 1 and 3 DF, p-value: 0.009378

a.

13.6679

0.1396

b. Standard errors of the estimates:

SE () = 0.6149

c.

SE () = 0.0234

d.

Hypothesis:

H₀: = 0

H₁ : 0

P value = 0.009

Conclusion:

Reject under H0 and conclude that a linear relation exist between height and head circumference at the level of significance alpha = 0.01.

e. Confidence interval:

> confint(fit)
2.5 % 97.5 %
(Intercept) 11.71092027 15.6249288
x 0.06526878 0.2139765

95% confidence interval for slope is

lower bound = 0.065

upper bound = 0.214

f.

height = x= 26.5 then head circumference is calculated as follows:

head circumference = 13.6679 + 0.1396 * 26.5

= 17.3673

head circumference = 17.37