Question

use the table to fill in the blanks. round to the nearest hundredths. website advertisement number of visits to website, x number of clicks on advertisement, y 153 629 471 914 307 1,045 510 1,106 14 38 30 53 21 60 32 63 correlation coefficient type your answer make a prediction of the number of clicks on the advertisement if 1,500 people visit the website type your answer slope type your answer... y - intercept type your answer... equation y = type your answer... x + type your answer...

Explanation:

Step1: Calculate the means of \(x\) and \(y\)

Let \(x\) be the number of website visits and \(y\) be the number of clicks on advertisement.
\(\bar{x}=\frac{153 + 629+471+914+307+1045+510+1106}{8}=\frac{5135}{8}=641.875\)
\(\bar{y}=\frac{14 + 38+30+53+21+60+32+63}{8}=\frac{311}{8}=38.875\)

Step2: Calculate the slope \(m\)

The formula for the slope \(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=(153 - 641.875)(14 - 38.875)+(629 - 641.875)(38 - 38.875)+(471-641.875)(30 - 38.875)+(914 - 641.875)(53 - 38.875)+(307 - 641.875)(21 - 38.875)+(1045 - 641.875)(60 - 38.875)+(510 - 641.875)(32 - 38.875)+(1106 - 641.875)(63 - 38.875)\)
\(=(- 488.875)(-24.875)+(-12.875)(- 0.875)+(-170.875)(-8.875)+(272.125)(14.125)+(-334.875)(-17.875)+(403.125)(21.125)+(-131.875)(-6.875)+(464.125)(24.125)\)
\(=12160.703125 + 11.15625+1516.796875+3843.046875+5984.140625+8514.046875+907.03125+11195.546875\)
\(=44132.475\)

\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=(153 - 641.875)^{2}+(629 - 641.875)^{2}+(471 - 641.875)^{2}+(914 - 641.875)^{2}+(307 - 641.875)^{2}+(1045 - 641.875)^{2}+(510 - 641.875)^{2}+(1106 - 641.875)^{2}\)
\(=(-488.875)^{2}+(-12.875)^{2}+(-170.875)^{2}+(272.125)^{2}+(-334.875)^{2}+(403.125)^{2}+(-131.875)^{2}+(464.125)^{2}\)
\(=239098.265625+165.765625+29198.140625+74050.140625+112140.140625+162491.265625+17391.421875+215412.140625\)
\(=850917.34375\)

\(m=\frac{44132.475}{850917.34375}\approx0.05\)

Step3: Calculate the \(y -\)intercept \(b\)

Using the formula \(y=mx + b\), substitute \(m = 0.05\), \(x=\bar{x}=641.875\) and \(y=\bar{y}=38.875\)
\(38.875=0.05\times641.875 + b\)
\(38.875 = 32.09375+b\)
\(b=38.875 - 32.09375=6.78\)

The regression equation is \(y = 0.05x+6.78\)

Step4: Calculate the correlation coefficient \(r\)

The formula for the correlation coefficient \(r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}\)
First, calculate \(\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}=(14 - 38.875)^{2}+(38 - 38.875)^{2}+(30 - 38.875)^{2}+(53 - 38.875)^{2}+(21 - 38.875)^{2}+(60 - 38.875)^{2}+(32 - 38.875)^{2}+(63 - 38.875)^{2}\)
\(=(-24.875)^{2}+(-0.875)^{2}+(-8.875)^{2}+(14.125)^{2}+(-17.875)^{2}+(21.125)^{2}+(-6.875)^{2}+(24.125)^{2}\)
\(=618.765625+0.765625+78.765625+199.515625+319.515625+446.265625+47.265625+582.015625\)
\(=2292.875\)

\(r=\frac{44132.475}{\sqrt{850917.34375\times2292.875}}\)
\(=\frac{44132.475}{\sqrt{1.951977\times10^{9}}}\)
\(=\frac{44132.475}{44181.19}\approx0.999\approx1.00\)

Step5: Make a prediction

When \(x = 1500\)
\(y=0.05\times1500+6.78=75 + 6.78=81.78\)

Answer:

Slope: \(0.05\)
\(y -\)intercept: \(6.78\)
Equation \(y=0.05x + 6.78\)
Correlation Coefficient: \(1.00\)
Prediction when \(x = 1500\): \(81.78\)