Question

the table below shows the time, in hours, spent by students on electronic devices and their math test scores. the data collected model a linear regression.
time spent on an electronic device (hours) | math test score
3 | 85
1 | 99
4 | 81
0 | 98
3 | 90
7 | 65
5 | 78
2 | 90
what is the correlation coefficient, to the nearest hundredth, for these data?
0.98
0.95

• 0.95
• 0.98

Explanation:

Step1: Calculate means

Let \(x\) be the time - spent on electronic devices and \(y\) be the math test scores.
\(n = 8\)
\(\bar{x}=\frac{3 + 1+4 + 0+3 + 7+5 + 2}{8}=\frac{25}{8}=3.125\)
\(\bar{y}=\frac{85 + 99+81 + 98+90 + 65+78 + 90}{8}=\frac{696}{8}=87\)

Step2: Calculate numerator and denominator components

\(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=(3 - 3.125)(85 - 87)+(1 - 3.125)(99 - 87)+(4 - 3.125)(81 - 87)+(0 - 3.125)(98 - 87)+(3 - 3.125)(90 - 87)+(7 - 3.125)(65 - 87)+(5 - 3.125)(78 - 87)+(2 - 3.125)(90 - 87)\)
\(=(- 0.125)\times(-2)+(-2.125)\times12+(0.875)\times(-6)+(-3.125)\times11+(-0.125)\times3+(3.875)\times(-22)+(1.875)\times(-9)+(-1.125)\times3\)
\(=0.25-25.5 - 5.25-34.375 - 0.375-85.25-16.875 - 3.375=-169\)

\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=(3 - 3.125)^{2}+(1 - 3.125)^{2}+(4 - 3.125)^{2}+(0 - 3.125)^{2}+(3 - 3.125)^{2}+(7 - 3.125)^{2}+(5 - 3.125)^{2}+(2 - 3.125)^{2}\)
\(=(-0.125)^{2}+(-2.125)^{2}+(0.875)^{2}+(-3.125)^{2}+(-0.125)^{2}+(3.875)^{2}+(1.875)^{2}+(-1.125)^{2}\)
\(=0.015625 + 4.515625+0.765625 + 9.765625+0.015625 + 15.015625+3.515625 + 1.265625 = 34.875\)

\(\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}=(85 - 87)^{2}+(99 - 87)^{2}+(81 - 87)^{2}+(98 - 87)^{2}+(90 - 87)^{2}+(65 - 87)^{2}+(78 - 87)^{2}+(90 - 87)^{2}\)
\(=(-2)^{2}+12^{2}+(-6)^{2}+11^{2}+3^{2}+(-22)^{2}+(-9)^{2}+3^{2}\)
\(=4 + 144+36 + 121+9+484+81+9 = 888\)

\(r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}=\frac{-169}{\sqrt{34.875\times888}}=\frac{-169}{\sqrt{30919.5}}\approx\frac{-169}{175.84}\approx - 0.96\approx - 0.95\)

Answer:

• 0.95