find the equation of the regression line for the given data. then construct a scatter - plot of the data and…

find the equation of the regression line for the given data. then construct a scatter - plot of the data and draw the regression line. (the pair of variables have a significant correlation.) then use the regression equation to predict the value of y for each of the given x - values, if meaningful. the table below shows the heights (in feet) and the number of stories of six notable buildings in a city. \n|height, x|774|625|521|508|497|477|\n|stories, y|51|47|45|42|38|35|\n(a) x = 498 feet\n(b) x = 630 feet\n(c) x = 315 feet\n(d) x = 728 feet\nfind the regression equation.\n\\(\\hat{y}=\\square x+\\square\\)\n(round the slope to three decimal places as needed. round the y - intercept to two decimal places as needed.)

find the equation of the regression line for the given data. then construct a scatter - plot of the data and draw the regression line. (the pair of variables have a significant correlation.) then use the regression equation to predict the value of y for each of the given x - values, if meaningful. the table below shows the heights (in feet) and the number of stories of six notable buildings in a city. \n|height, x|774|625|521|508|497|477|\n|stories, y|51|47|45|42|38|35|\n(a) x = 498 feet\n(b) x = 630 feet\n(c) x = 315 feet\n(d) x = 728 feet\nfind the regression equation.\n\\(\\hat{y}=\\square x+\\square\\)\n(round the slope to three decimal places as needed. round the y - intercept to two decimal places as needed.)

Answer

Explanation:

Step1: Calculate necessary sums

Let (n = 6). We need to calculate (\sum x), (\sum y), (\sum xy), (\sum x^{2}). (\sum x=774 + 626+521+508+497+477 = 3403) (\sum y=51 + 47+45+42+38+35 = 258) (\sum xy=(774\times51)+(626\times47)+(521\times45)+(508\times42)+(497\times38)+(477\times35)) (=39474+29422+23445+21336+18886+16695 = 149258) (\sum x^{2}=774^{2}+626^{2}+521^{2}+508^{2}+497^{2}+477^{2}) (=599076+391876+271441+258064+247009+227529 = 1994995)

Step2: Calculate the slope (b_1)

The formula for the slope (b_1=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}) [ \begin{align*} b_1&=\frac{6\times149258 - 3403\times258}{6\times1994995-(3403)^{2}}\ &=\frac{895548-877974}{11969970 - 11580409}\ &=\frac{17574}{389561}\ &\approx0.045 \end{align*} ]

Step3: Calculate the y - intercept (b_0)

The formula for the y - intercept (b_0=\bar{y}-b_1\bar{x}), where (\bar{x}=\frac{\sum x}{n}=\frac{3403}{6}\approx567.17) and (\bar{y}=\frac{\sum y}{n}=\frac{258}{6} = 43) [ \begin{align*} b_0&=43-0.045\times567.17\ &=43 - 25.52265\ &\approx17.48 \end{align*} ]

Answer:

(\hat{y}=0.045x + 17.48)