the table below shows the time, in hours, spent by students on electronic devices and their math test…

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.\ntime spent on an electronic device (hours) | math test score\n3 | 85\n1 | 99\n4 | 81\n0 | 98\n3 | 90\n7 | 65\n5 | 78\n2 | 90\nwhat is the correlation coefficient, to the nearest hundredth, for these data?\n0.98\n0.95\n- 0.95\n- 0.98

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.\ntime spent on an electronic device (hours) | math test score\n3 | 85\n1 | 99\n4 | 81\n0 | 98\n3 | 90\n7 | 65\n5 | 78\n2 | 90\nwhat is the correlation coefficient, to the nearest hundredth, for these data?\n0.98\n0.95\n- 0.95\n- 0.98

Answer

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