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 number of hours 6 students spent for a test and their scores on that test are shown below. hours spent studying, x 1 2 3 4 5 5 test score, y 40 44 50 49 62 67 (a) x = 2 hours (b) x = 3.5 hours (c) x = 12 hours (d) x = 1.5 hours find the regression equation (hat{y}=square x+square) (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 number of hours 6 students spent for a test and their scores on that test are shown below. hours spent studying, x 1 2 3 4 5 5 test score, y 40 44 50 49 62 67 (a) x = 2 hours (b) x = 3.5 hours (c) x = 12 hours (d) x = 1.5 hours find the regression equation (hat{y}=square x+square) (round the slope to three decimal places as needed. round the y - intercept to two decimal places as needed.)

Answer

Explanation:

Step1: Calculate sums

Let (n = 6). (\sum x=1 + 2+3 + 4+5+5=20) (\sum y=40 + 44+50+49+62+67 = 312) (\sum x^{2}=1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+5^{2}=1 + 4+9+16+25+25 = 80) (\sum xy=1\times40+2\times44 + 3\times50+4\times49+5\times62+5\times67=40+88+150+196+310+335 = 1119)

Step2: Calculate slope (m)

The formula for the slope (m) of the regression - line is (m=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}) [ \begin{align*} m&=\frac{6\times1119 - 20\times312}{6\times80-20^{2}}\ &=\frac{6714-6240}{480 - 400}\ &=\frac{474}{80}\ &=5.925 \end{align*} ]

Step3: Calculate y - intercept (b)

The formula for the y - intercept (b) is (b=\frac{\sum y-m\sum x}{n}) [ \begin{align*} b&=\frac{312-5.925\times20}{6}\ &=\frac{312 - 118.5}{6}\ &=\frac{193.5}{6}\ &=32.25 \end{align*} ]

The regression equation is (\hat{y}=5.925x + 32.25)

Step4: Predict values

(a) When (x = 2)

(\hat{y}=5.925\times2+32.25=11.85+32.25 = 44.1)

(b) When (x = 3.5)

(\hat{y}=5.925\times3.5+32.25=20.7375+32.25 = 52.9875)

(c) When (x = 12)

Since (x = 12) is outside the range of the data ((1\leq x\leq5)), the prediction may not be meaningful. But using the equation (\hat{y}=5.925\times12+32.25=71.1+32.25 = 103.35)

(d) When (x = 1.5)

(\hat{y}=5.925\times1.5+32.25=8.8875+32.25 = 41.1375)

Answer:

The regression equation is (\hat{y}=5.925x + 32.25) (a) (44.1) (b) (52.9875) (c) (103.35) (may not be meaningful as (x) is outside the data - range) (d) (41.1375)