Question

The accompanying table shows the ages​ (in years) of

11

children and the numbers of words in their vocabulary. Complete parts​ (a) through​ (d) below.

...

​(a) Display the data in a scatter plot. Choose the correct graph below.

A.

0246806001200180024003000Age (years)Vocabulary size

A scatter plot has a horizontal axis labeled Age in years from 0 to 8 in increments of 1 and a vertical axis labeled Vocabulary size from 0 to 3000 in increments of 300. The following 11 points are plotted: (1, 0); (2, 250); (2, 250); (3, 550); (3, 550); (4, 1200); (4, 1300); (5, 2200); (5, 2400); (6, 2400); (6, 2500). From left to right, the points follow a general trend of rising from left to right at a constant rate. All vertical coordinates are approximate.

B.

01500300002468Age (years)Vocabulary size

A scatter plot has a horizontal axis labeled Age in years from 0 to 3000 in increments of 300 and a vertical axis labeled Vocabulary size from 0 to 8 in increments of 1. The following 11 points are plotted: (0, 6); (250, 6); (250, 5); (550, 5); (550, 4); (1200, 4); (1300, 3); (2200, 3); (2400, 2); (2400, 2); (2500, 1). The points follow a general trend of falling from left to right at a constant rate. All horizontal coordinates are approximate.

C.

01500300002468Age (years)Vocabulary size

A scatter plot has a horizontal axis labeled Age in years from 0 to 3000 in increments of 300 and a vertical axis labeled Vocabulary size from 0 to 8 in increments of 1. The following 11 points are plotted: (0, 1); (250, 2); (250, 2); (550, 3); (550, 3); (1200, 4); (1300, 4); (2200, 5); (2400, 5); (2400, 6); (2500, 6). The points follow a general trend of rising from left to right at a constant rate. All horizontal coordinates are approximate.

D.

0246806001200180024003000Age (years)Vocabulary size

A scatter plot has a horizontal axis labeled Age in years from 0 to 8 in increments of 1 and a vertical axis labeled Vocabulary size from 0 to 3000 in increments of 300. The following 11 points are plotted: (1, 2500); (2, 2400); (2, 2400); (3, 2200); (3, 1300); (4, 1200); (4, 550); (5, 550); (5, 250); (6, 250); (6, 0). The points follow a general trend of falling from left to right at a constant rate. All vertical coordinates are approximate.

​(b) Calculate the sample correlation coefficient r.

requals=nothing

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

​(c) Describe the type of​ correlation, if​ any, and interpret the correlation in the context of the data.

There is

▼

a strong negative

a perfect negative

no

a strong positive

a weak negative

a perfect positive

a weak positive

linear correlation.

Interpret the correlation. Choose the correct answer below.

A.

As age​ increases, the number of words in​ children's vocabulary tends to decrease.

B.

Based on the​ correlation, there does not appear to be any relationship between​ children's ages and the number of words in their vocabulary.

C.

As age​ increases, the number of words in​ children's vocabulary tends to increase.

D.

Aging causes the number of words in​ children's vocabulary to increase.

E.

Based on the​ correlation, there does not appear to be a linear relationship between​ children's ages and the number of words in their vocabulary

F.

Aging causes the number of words in​ children's vocabulary to decrease.

​(d) Use the table of critical values for the Pearson correlation coefficient to make a conclusion about the correlation coefficient. Let

alphaαequals=0.010.01.

The critical value is

nothing.

​Therefore, there

▼

is not

is

sufficient evidence at the

11​%

level of significance to conclude that

▼

there is a significant linear correlation

there is no correlation

between​ children's ages and the number of words in their vocabulary.

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

​Age, x	Vocabulary​ size, y
1	7
2	270
3	530
4	1200
5	2200
6	2500
3	550
5	2400
2	240
4	1300
6	2400

Answer 1

Following is the scatter plot of the data:

(b)

Following table shows the calculations:

	X	Y	X^2	Y^2	XY
	1	7	1	49	7
	2	270	4	72900	540
	3	530	9	280900	1590
	4	1200	16	1440000	4800
	5	2200	25	4840000	11000
	6	2500	36	6250000	15000
	3	550	9	302500	1650
	5	2400	25	5760000	12000
	2	240	4	57600	480
	4	1300	16	1690000	5200
	6	2400	36	5760000	14400
Total	41	13597	181	26453949	66667

Answer: r = 0.970

(c)

a strong positive

C As age​ increases, the number of words in​ children's vocabulary tends to increase.

(d)

The critical value of r from the table is 0.684

Since r > 0.684 so we can conclude that relationship between the variables is significant.

​Therefore, there s sufficient evidence at the 1​% level of significance to conclude that there is a significant linear correlation between​ children's ages and the number of words in their vocabulary.