Question

A pediatrician wants to determine the relation that may exist between a​ child's height and head circumference. She randomly selects 8 children from her​ practice, measures their height and head​ circumference, and obtains the data shown in the table. Complete parts​ (a) through​ (e) to the right. Height​ (in.) Head Circumference​ (in.) 27.75 17.4 25 17.1 26.25 17.3 25.75 17 27.5 17.4 26.25 17.2 26 17.1 26.75 17.4 LOADING... Click here to see the Table of Critical Values for Correlation Coefficient. ​(a) If the pediatrician wants to use height to predict head​ circumference, determine which variable is the explanatory variable and which is the response variable. The explanatory variable is head circumference and the response variable is height. The explanatory variable is height and the response variable is head circumference. Your answer is correct. ​(b) Draw a scatter diagram. Which of the following represents the​ data? A. 16.917.6 2528Height (in)Circ. (in) A scatter diagram has a horizontal axis labeled “Height (inches)” from 16.9 to 17.6 in increments of 0.1 and a vertical axis labeled “Circumference (inches)” from 25 to 28 in increments of 0.5. The following 8 points are plotted, listed here from left to right: (17, 25.8); (17.1, 25); (17.1, 26); (17.2, 26.3); (17.3, 26.3); (17.4, 26.8); (17.4, 27.5); (17.4, 27.8). All vertical coordinates are approximate. The points generally rise from left to right at a constant rate. B. 2528 16.917.6Circ. (in)Height (in) A scatter diagram has a horizontal axis labeled “Circumference (inches)” from 25 to 28 in increments of 0.5 and a vertical axis labeled “Height (inches)” from 16.9 to 17.6 in increments of 0.1. The following 8 points are plotted, listed here from left to right: (25, 17.1); (25.8, 17); (26, 17.1); (26.3, 17.2); (26.3, 17.3); (26.8, 17.4); (27.5, 17.4); (27.8, 17.4). All horizontal coordinates are approximate. The points generally rise from left to right at a constant rate. C. 16.917.6 2528Circ. (in)Height (in) A scatter diagram has a horizontal axis labeled “Circumference (inches)” from 16.9 to 17.6 in increments of 0.1 and a vertical axis labeled “Height (inches)” from 25 to 28 in increments of 0.5. The following 8 points are plotted, listed here from left to right: (17, 25.8); (17.1, 25); (17.1, 26); (17.2, 26.3); (17.3, 26.3); (17.4, 26.8); (17.4, 27.5); (17.4, 27.8). All vertical coordinates are approximate. The points generally rise from left to right at a constant rate. D. 2528 16.917.6Height (in)Circ. (in) A scatter diagram has a horizontal axis labeled “Height (inches)” from 25 to 28 in increments of 0.5 and a vertical axis labeled “Circumference (inches)” from 16.9 to 17.6 in increments of 0.1. The following 8 points are plotted, listed here from left to right: (25, 17.1); (25.8, 17); (26, 17.1); (26.3, 17.2); (26.3, 17.3); (26.8, 17.4); (27.5, 17.4); (27.8, 17.4). All horizontal coordinates are approximate. The points generally rise from left to right at a constant rate. Your answer is correct. ​(c) Compute the linear correlation coefficient between the height and head circumference of a child. requals 0.842 ​(Round to three decimal places as​ needed.) ​(d) Does a linear relation exist between height and head​ circumference? ​(Round to three decimal places as​ needed.)

A. ​Yes, the variables height and head circumference are positively associated because r is negative and the absolute value of the correlation coefficient is greater than the critical​ value, nothing .

B. ​No, the variables height and head circumference are not linearly related because r is positive and the absolute value of the correlation coefficient is less than the critical​ value, nothing
C.No, the variables height and head circumference are not linearly related because r is negative and the absolute value of the correlation coefficient is less than the critical​ value, nothing . D. ​Yes, the variables height and head circumference are positively associated because r is positive and the absolute value of the correlation coefficient is greater than the critical​ value, nothing

Critical Values for Correlation Coefficient

n 3 0.997 4 0.950 5 0.878 6 0.811 7 0.754 8 0.707 9 0.666 10 0.632 11 0.602 12 0.576 13 0.553 14 0.532 15 0.514 16 0.497 17 0.482 18 0.468 19 0.456 20 0.444 21 0.433 22 0.423 23 0.413 24 0.404 25 0.396 26 0.388 27 0.381 28 0.374 29 0.367 30 0.361 n

Answer 1

(a)

Height is used to predict head​ circumference.

So, height is independent variable and head​ circumference is dependent variable. Thus head​ circumference can be calculated (and so explained) based on height.

Hence, the explanatory variable is height and the response variable is head circumference.

(b)

The scatter diagram is as follows.

From the diagram, we observe that-

• A scatter diagram has a horizontal axis labeled “Height (inches)” from 25 to 28 in increments of 0.5.
• It has a vertical axis labeled “Circumference (inches)” from 16.9 to 17.6 in increments of 0.1.
• The following 8 points are plotted, listed here from left to right as (25, 17.1); (25.8, 17); (26, 17.1); (26.3, 17.2); (26.3, 17.3); (26.8, 17.4); (27.5, 17.4); (27.8, 17.4). All horizontal coordinates are approximate.
• The points generally rise from left to right at a constant rate.

So, (D) is correct.

(c)

Correlation coefficient is given by

(d)

We have to test for null hypothesis

against the alternative hypothesis

Sample size

Degrees of freedom

From the given table,

Also, we have

We reject null hypothesis if .

Here we observe that .

So, we reject null hypothesis.

Hence based on the given data, we conclude that

D. ​Yes, the variables height and head circumference are positively associated because r is positive and the absolute value of the correlation coefficient is greater than the critical​ value.

So, (D) is correct.