Question

the accompanying table shows the height (in inches) of 8 high school girls and their scores on an iq test. complete parts (a) through (c) below. (a) display the data in a scatter - plot. choose the correct graph below. (b) calculate the sample correlation coefficient r. r = □ (round to three decimal places as needed.) (c) describe the type of correlation, if any, and interpret the correlation in the there is a strong positive linear correlation. data table: height, x: 62, 57, 63, 67, 57, 63, 63, 55; iq score, y: 107, 99, 106, 111, 94, 108, 118, 126

Explanation:

Step1: Recall scatter - plot concept

The x - axis represents height and y - axis represents IQ score. Plot the points \((x,y)\) from the data table.

Step2: Calculate the sample correlation coefficient formula

The formula for the sample correlation coefficient \(r\) is \(r=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{\sqrt{[n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}][n\sum_{i = 1}^{n}y_i^{2}-(\sum_{i = 1}^{n}y_i)^{2}]}}\)
Let \(n = 8\).
First, calculate \(\sum_{i=1}^{8}x_i=62 + 57+63+67+57+63+63+55=487\)
\(\sum_{i = 1}^{8}y_i=107 + 99+106+111+94+108+118+126=869\)
\(\sum_{i=1}^{8}x_i^{2}=62^{2}+57^{2}+63^{2}+67^{2}+57^{2}+63^{2}+63^{2}+55^{2}\)
\(=3844 + 3249+3969+4489+3249+3969+3969+3025 = 29763\)
\(\sum_{i=1}^{8}y_i^{2}=107^{2}+99^{2}+106^{2}+111^{2}+94^{2}+108^{2}+118^{2}+126^{2}\)
\(=11449+9801+11236+12321+8836+11664+13924+15876=95007\)
\(\sum_{i=1}^{8}x_iy_i=62\times107+57\times99+63\times106+67\times111+57\times94+63\times108+63\times118+55\times126\)
\(=6634+5643+6678+7437+5358+6804+7434+6930 = 52528\)

Substitute these values into the formula:
\[

$$\begin{align*} r&=\frac{8\times52528-487\times869}{\sqrt{[8\times29763-(487)^{2}][8\times95007-(869)^{2}]}}\\ &=\frac{420224 - 423103}{\sqrt{[238104 - 237169][760056-755161]}}\\ &=\frac{- 2879}{\sqrt{(935)(4895)}}\\ &=\frac{-2879}{\sqrt{4576825}}\\ &=\frac{-2879}{2139.352}\\ &\approx - 0.135 \end{align*}$$

\]

Step3: Interpret the correlation

Since \(r\approx - 0.135\), the absolute value of \(r\) is close to \(0\). So there is a weak negative linear correlation.

Answer:

(a) Without seeing all the options clearly, we would plot points \((62,107),(57,99),(63,106),(67,111),(57,94),(63,108),(63,118),(55,126)\) on a scatter - plot where x - axis is height and y - axis is IQ score.
(b) \(r\approx - 0.135\)
(c) There is a weak negative linear correlation. The correlation indicates that as height increases, the IQ score has a very slight tendency to decrease, but the relationship is very weak.