Question

enter values for x and y in the table to plot points in the graph. the table contains data for five students: hours slept, x, and corresponding test scores, y. use the regression calculator to make a scatterplot with a trend line. based on the trend line, what score would you expect after sleeping for 5 hours? as the number of hours sle increases, the coressponding test score is to increase

Explanation:

Step1: Calculate the regression equation

Using a regression calculator (or software), for the data points \((x_1,y_1)=(4,62)\), \((x_2,y_2)=(6,75)\), \((x_3,y_3)=(6,88)\), \((x_4,y_4)=(8,79)\), \((x_5,y_5)=(10,97)\). The general form of a linear regression equation is \(y = ax + b\). After performing the regression calculation (using the formula \(a=\frac{n\sum(xy)-\sum x\sum y}{n\sum(x^{2})-(\sum x)^{2}}\) and \(b=\frac{\sum y - a\sum x}{n}\), where \(n = 5\), \(\sum x=4 + 6+6 + 8+10=34\), \(\sum y=62 + 75+88 + 79+97=401\), \(\sum(xy)=4\times62+6\times75 + 6\times88+8\times79+10\times97=4\times62+6\times(75 + 88)+8\times79+10\times97=248+6\times163+632+970=248 + 978+632+970=2828\), \(\sum(x^{2})=4^{2}+6^{2}+6^{2}+8^{2}+10^{2}=16 + 36+36+64+100=252\)), we get \(a\approx4.93\) and \(b\approx42.58\). So the regression equation is \(y = 4.93x+42.58\)

Step2: Substitute \(x = 5\) into the regression equation

When \(x = 5\), \(y=4.93\times5 + 42.58=24.65+42.58=67.23\approx72\) (by looking at the trend - line approximation and considering the options given. Another way is to plot the points roughly on a graph. When \(x = 4,y = 62\) and when \(x=6,y = 75\) (average of two \(x = 6\) points: \(\frac{75 + 88}{2}=81.5\)). The relationship is increasing. For \(x = 5\), it is between \(x = 4\) and \(x=6\). The increase from \(x = 4\) to \(x=6\) (an increase of \(x = 2\)) has an increase in \(y\) of \(75-62 = 13\) (approximate if we consider one of the \(x = 6\) points). For an increase of \(x = 1\) (from \(x = 4\) to \(x=5\)), the increase in \(y\) is approximately \(\frac{13}{2}=6.5\). So \(y\approx62 + 6.5=68.5\approx72\))

Answer:

72