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 26.75 25.5 27.75 26.5 27.5 Head Circumference​ (inches), y 17.3 17.1 17.6 17.3 17.5 a) Treating height as the explanatory? variable, x, use technology to determine the estimates of ?0 and ?1. (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 ?=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

a) Estimated beta_0=11.57795 ; Estimated beta_1=0.21575

b) SE(beta_0)=0.56386 ; SE(beta_1)=0.02103

c) Using technology the residual variance is 0.001404001, R square is 97.23%

(For this part, the mathematical symbols in the uploaded question is not properly encoded. If you comment with the correct need of the question , I will reply with the needed answer)

d) H₀: beta_1=0 against H₁: beta_1 is non zero

p value of the test (based on a t statistic) is .001975

Since the p value is less than the nominal value of 1%, evidence is strong against the null and we reject the null. Hence, significant linear relation between the variables x & y exists.

e) 95% CI for beta_0 is ( 9.7835071 ,13.3723984)

95% CI for beta_1 is (  0.1488207, 0.2826754)

f) A good guess is the predicted value obtained from regression , which is 11.57795+26.5*0.21575
=17.29532 inches