Question

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 27 25.5 27.75 25 26.5 Head Circumference​ (inches), y 17.5 17.1 17.6 16.9 17.3 ​(a) Treating height as the explanatory​ variable, x, use technology to determine the estimates of beta 0 and beta 1. beta 0 almost equalsb 0 equals ???????? ​(Round to four decimal places as​ needed.) beta 1almost equalsb 1equals nothing ​(Round to four decimal places as​ needed.) (b) Use technology to compute the standard error of the​ estimate, s Subscript e. s Subscript eequals ????​(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 1. s Subscript b 1equals ??????   ​(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 alphaequals0.01 level of significance. State the null and alternative hypotheses for this test. Choose the correct answer below. A. Upper H 0​: beta 0equals0 Upper H 1​: beta 0not equals0 B. Upper H 0​: beta 0equals0 Upper H 1​: beta 0greater than0 C. Upper H 0​: beta 1equals0 Upper H 1​: beta 1not equals0 Your answer is correct.D. Upper H 0​: beta 1equals0 Upper H 1​: beta 1greater than0 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. Do not reject Upper H 0 and conclude that a linear relation does not exist between a​ child's height and head circumference at the level of significance alphaequals0.01. B. Reject Upper H 0 and conclude that a linear relation does not exist between a​ child's height and head circumference at the level of significance alphaequals0.01. C. Do not reject Upper H 0 and conclude that a linear relation exists between a​ child's height and head circumference at the level of significance alphaequals0.01. D. Reject Upper H 0 and conclude that a linear relation exists between a​ child's height and head circumference at the level of significance alphaequals0.01. ​(e) Use technology to construct a​ 95% confidence interval about the slope of the true​ least-squares regression line. Lower​ bound: ???? Upper​ bound: ????? ​(Round to three decimal 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 estimate of the​ child's head circumference would be ???????? ​(Round to two decimal places as​ needed.)

Answer 1

I used r software to solve this problem,

R codes

x=c(27,25.5,27.75,25,26.5)
y=c(17.5,17.1,17.6,16.9,17.3)
fit=lm(y~x)
summary(fit)

Call:
lm(formula = y ~ x)

Residuals:
1 2 3 4 5
0.05455 0.03636 -0.03636 -0.03636 -0.01818

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 10.57273 0.58355 18.12 0.000367 ***
x 0.25455 0.02213 11.50 0.001411 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.04924 on 3 degrees of freedom
Multiple R-squared: 0.9778, Adjusted R-squared: 0.9704
F-statistic: 132.3 on 1 and 3 DF, p-value: 0.001411

> confint(fit)
2.5 % 97.5 %
(Intercept) 8.7156220 12.4298326
x 0.1841173 0.3249736