Answer to Question #346245 in Statistics and Probability for wiwili

Question #346245

A study was conducted to investigate the relationship between roadway surface temperature (x) and pavement deflection (y). The data are as follows:


x

65.3

66.4

70.8

70.7

69.8

71.6

72.1

72.4

73.5


y

0.6144

0.6273

0.6668

0.6781

0.6497

0.6678

0.6783

0.6755

0.6873


Calculate the Pearson correlation coefficient. Round your answers to the nearest hundredths.


1
Expert's answer
2022-05-31T10:58:20-0400

The following table needs to be used:


XYXYX2Y265.30.614440.1203244264.090.3774873666.40.627341.652724408.960.3935052970.80.666847.209445012.640.4446222470.70.678147.941674998.490.4598196169.80.649745.349064872.040.4221100971.60.667847.814485126.560.4459568472.10.678348.905435198.410.4600908972.40.675548.90625241.760.4563002573.50.687350.516555402.250.47238129Sum=632.65.9452418.4158744525.23.93227386\def\arraystretch{1.5} \begin{array}{c:c:c:c:c} X & Y & XY & X^2 & Y^2 \\ \hline 65.3 & 0.6144 & 40.12032 & 44264.09 & 0.37748736 \\ \hdashline 66.4 & 0.6273 & 41.65272 & 4408.96 & 0.39350529 \\ \hdashline 70.8 & 0.6668 & 47.20944 & 5012.64 & 0.44462224 \\ \hdashline 70.7 & 0.6781 & 47.94167 & 4998.49 & 0.45981961 \\ \hdashline 69.8 & 0.6497 & 45.34906 & 4872.04 & 0.42211009 \\ \hdashline 71.6 & 0.6678 & 47.81448 & 5126.56 & 0.44595684 \\ \hdashline 72.1 & 0.6783 & 48.90543 & 5198.41 & 0.46009089 \\ \hdashline 72.4 & 0.6755 & 48.9062 & 5241.76 & 0.45630025 \\ \hdashline 73.5 & 0.6873 & 50.51655 & 5402.25 & 0.47238129 \\ \hdashline Sum =\\ 632.6 & 5.9452 & 418.41587 & 44525.2 & 3.93227386\\ \hdashline \end{array}




Xˉ=1niXi=632.69\bar{X}=\dfrac{1}{n}\sum _{i}X_i=\dfrac{632.6}{9}=70.288889=70.288889Yˉ=1niYi=5.94529\bar{Y}=\dfrac{1}{n}\sum _{i}Y_i=\dfrac{5.9452}{9}=0.660578=0.660578SSXX=iXi21n(iXi)2SS_{XX}=\sum_iX_i^2-\dfrac{1}{n}(\sum _{i}X_i)^2=44525.2632.629=60.448889=44525.2-\dfrac{632.6^2}{9}=60.448889SSYY=iYi21n(iYi)2SS_{YY}=\sum_iY_i^2-\dfrac{1}{n}(\sum _{i}Y_i)^2=3.932273865.945229=0.005007=3.93227386-\dfrac{5.9452^2}{9}=0.005007




SSXY=iXiYi1n(iXi)(iYi)SS_{XY}=\sum_iX_iY_i-\dfrac{1}{n}(\sum _{i}X_i)(\sum _{i}Y_i)=418.41587632.6(5.9452)9=0.534368=418.41587-\dfrac{632.6(5.9452)}{9}=0.534368r=SSXYSSXXSSYY=0.53436860.448889(0.005007)r=\dfrac{SS_{XY}}{\sqrt{SS_{XX}SS_{YY}}}=\dfrac{0.534368}{\sqrt{60.448889(0.005007)}}




=0.97132277=0.97132277


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