Question

the accompanying table shows the ages (in years) of 11 children and the numbers of words in their vocabulary. complete parts (a) through (c) below. click here to view the data table. (a) display the data in a scatter - plot. choose the correct graph below. (b) calculate the sample correlation coefficient r. (round to three decimal places as needed.) (c) describe the type of correlation, if any, and interpret the correlation in the context of the there is linear correlation. data table age, x vocabulary size, y 1 8 2 270 3 500 4 1100 5 700 6 2600 3 720 8 2300 2 200 4 1300 6 2500

Explanation:

Step1: Calculate necessary sums

Let \(n = 11\). Calculate \(\sum_{i = 1}^{n}x_i\), \(\sum_{i = 1}^{n}y_i\), \(\sum_{i = 1}^{n}x_i^2\), \(\sum_{i = 1}^{n}y_i^2\) and \(\sum_{i = 1}^{n}x_iy_i\) from the data table.
\(\sum_{i = 1}^{11}x_i=1 + 2+3 + 4+5+6+3+8+2+4+6=44\)
\(\sum_{i = 1}^{11}y_i=8 + 270+500+1100+700+2600+720+2300+200+1300+2500 = 13198\)
\(\sum_{i = 1}^{11}x_i^2=1^2+2^2 + 3^2+4^2+5^2+6^2+3^2+8^2+2^2+4^2+6^2=1 + 4+9+16+25+36+9+64+4+16+36 = 220\)
\(\sum_{i = 1}^{11}y_i^2=8^2+270^2+500^2+1100^2+700^2+2600^2+720^2+2300^2+200^2+1300^2+2500^2\)
\(=64 + 72900+250000+1210000+490000+6760000+518400+5290000+40000+1690000+6250000=22579364\)
\(\sum_{i = 1}^{11}x_iy_i=1\times8+2\times270 + 3\times500+4\times1100+5\times700+6\times2600+3\times720+8\times2300+2\times200+4\times1300+6\times2500\)
\(=8+540+1500+4400+3500+15600+2160+18400+400+5200+15000=66708\)

Step2: Use the correlation - coefficient formula

The sample correlation coefficient \(r\) is given by the formula:
\[r=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{\sqrt{n\sum_{i = 1}^{n}x_i^2-(\sum_{i = 1}^{n}x_i)^2}\sqrt{n\sum_{i = 1}^{n}y_i^2-(\sum_{i = 1}^{n}y_i)^2}}\]
Substitute \(n = 11\), \(\sum_{i = 1}^{11}x_i = 44\), \(\sum_{i = 1}^{11}y_i=13198\), \(\sum_{i = 1}^{11}x_i^2 = 220\), \(\sum_{i = 1}^{11}y_i^2=22579364\) and \(\sum_{i = 1}^{11}x_iy_i=66708\) into the formula.
\[n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i=11\times66708-44\times13198\]
\[=733788-580712=153076\]
\[n\sum_{i = 1}^{n}x_i^2-(\sum_{i = 1}^{n}x_i)^2=11\times220 - 44^2=2420-1936 = 484\]
\[n\sum_{i = 1}^{n}y_i^2-(\sum_{i = 1}^{n}y_i)^2=11\times22579364-13198^2\]
\[=248373004-174187204=74185800\]
\(\sqrt{n\sum_{i = 1}^{n}x_i^2-(\sum_{i = 1}^{n}x_i)^2}\sqrt{n\sum_{i = 1}^{n}y_i^2-(\sum_{i = 1}^{n}y_i)^2}=\sqrt{484}\sqrt{74185800}=22\times8613.12 = 189488.64\)
\[r=\frac{153076}{189488.64}\approx0.81\]

Step3: Determine the type of correlation

Since \(r\approx0.81\) and \(0 < r<1\), there is a strong positive linear correlation. This means that as the age of the children increases, the size of their vocabulary tends to increase.

Answer:

(b) \(r\approx0.81\)
(c) There is a strong positive linear correlation. As the age of children increases, the size of their vocabulary tends to increase.