Question

making a prediction using the regression calculator
enter values for x and y in the table to plot points in the graph. enter the data representing the weight loss plan where x is the weekly hours of aerobic activity and y is the pounds lost per month. use the regression equation to complete the statements.
(.25, 1), (1, 1.5), (1.2, 3), (2.25, 2.25), (2.5, 3.5), (2.8, 4), (3, 5.5), (3.5, 4.5), (4.5, 5.5), (5, 7)
a person who is active for 8 hours weekly could expect to lose pounds a month.
to lose 5 pounds a month, a person should plan to be active for hours a week.

Explanation:

Step1: Input data into regression calculator

Using the given data points \((0.25,1),(1,1.5),(1.2,3),(2.25,2.25),(2.5,3.5),(2.8,4),(3,5.5),(3.5,4.5),(4.5,5.5),(5,7)\) in a regression calculator (either online or in - built in statistical software).

Step2: Obtain regression equation

The regression equation of the form \(y = ax + b\) is found. After calculation (using the formula for the least - squares regression line \(a=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}\) and \(b=\overline{y}-a\overline{x}\), where \(n = 10\), \(\sum_{i=1}^{n}x_i=0.25 + 1+1.2 + 2.25+2.5+2.8+3+3.5+4.5+5=26.5\), \(\sum_{i = 1}^{n}y_i=1 + 1.5+3+2.25+3.5+4+5.5+4.5+5.5+7=38.25\), \(\sum_{i=1}^{n}x_i^{2}=0.25^{2}+1^{2}+1.2^{2}+2.25^{2}+2.5^{2}+2.8^{2}+3^{2}+3.5^{2}+4.5^{2}+5^{2}=0.0625 + 1+1.44+5.0625+6.25+7.84+9+12.25+20.25+25 = 88.15\), \(\sum_{i = 1}^{n}x_iy_i=(0.25\times1)+(1\times1.5)+(1.2\times3)+(2.25\times2.25)+(2.5\times3.5)+(2.8\times4)+(3\times5.5)+(3.5\times4.5)+(4.5\times5.5)+(5\times7)=0.25+1.5 + 3.6+5.0625+8.75+11.2+16.5+15.75+24.75+35=122.3625\)).

\(a=\frac{10\times122.3625-26.5\times38.25}{10\times88.15-(26.5)^{2}}=\frac{1223.625 - 1013.625}{881.5 - 702.25}=\frac{210}{179.25}\approx1.17\)

\(\overline{x}=\frac{26.5}{10}=2.65\), \(\overline{y}=\frac{38.25}{10}=3.825\)

\(b = 3.825-1.17\times2.65=3.825 - 3.1005 = 0.7245\)

The regression equation is \(y = 1.17x+0.7245\)

Step3: Predict for \(x = 8\)

Substitute \(x = 8\) into the regression equation \(y=1.17\times8 + 0.7245=9.36+0.7245 = 10.0845\approx10.1\)

Step4: Solve for \(x\) when \(y = 5\)

Set \(y = 5\) in the equation \(5=1.17x+0.7245\)

\(1.17x=5 - 0.7245=4.2755\)

\(x=\frac{4.2755}{1.17}\approx3.65\approx3.6\)

Answer:

A person who is active for 8 hours weekly could expect to lose \(10.1\) pounds a month. To lose 5 pounds a month, a person should plan to be active for \(3.6\) hours a week.