Question

frequent food questionnaires (ffq) are a simple way to obtain information on the foods individuals consume by asking them questions about typical amounts of food consumed in a day, a week, or a month. a more accurate picture is obtained by obtaining a detailed food diary (dr) for several days that are randomly chosen over a certain time period. the data obtained from a frequent food questionnaire can be compared with the food diary to assess the validity of the questionnaire. based on the seven individuals, what is/are true about the coefficient of determination value between food consumption from the food questionnaire and diary : table of values dr ffq 8.26 1.68 0.83 0 20.13 15.10 11.16 7.49 7.18 12.84 1.76 0 22.66 25.06 a. the coefficient of determination is 0.79 b. the coefficient of determination is. 0.89 c. since the relationship between the two variables is non - linear, r should be close to 1 d. all of the answer options are correct. e. 79% of the variability in y is explained by the regression of y on x

Explanation:

Step1: Recall coefficient of determination formula

The coefficient of determination $R^{2}$ measures the proportion of the variance in the dependent - variable that is predictable from the independent variable in a regression analysis. First, we would need to calculate the correlation coefficient $r$ between the two variables (DR and FFQ) using the formula $r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}$, and then $R^{2}=r^{2}$. But we can also use statistical software or a calculator with regression capabilities.
Let $x$ be the values from DR and $y$ be the values from FFQ.
$n = 7$
$\sum x=8.26 + 0.83+20.13+11.16+7.18+1.76+22.66=71.98$
$\sum y=1.68 + 0+15.10+7.49+12.84+0+25.06=62.17$
$\sum xy=(8.26\times1.68)+(0.83\times0)+(20.13\times15.10)+(11.16\times7.49)+(7.18\times12.84)+(1.76\times0)+(22.66\times25.06)$
$=13.8768+0 + 303.963+83.6884+92.2912+0+567.8596=1061.679$
$\sum x^{2}=8.26^{2}+0.83^{2}+20.13^{2}+11.16^{2}+7.18^{2}+1.76^{2}+22.66^{2}$
$=68.2276+0.6889+405.2169+124.5456+51.5524+3.0976+513.4756=1166.704$
$\sum y^{2}=1.68^{2}+0^{2}+15.10^{2}+7.49^{2}+12.84^{2}+0^{2}+25.06^{2}$
$=2.8224+0+228.01+56.1001+164.8656+0+628.0036=1079.8017$
$r=\frac{7\times1061.679 - 71.98\times62.17}{\sqrt{[7\times1166.704-(71.98)^{2}][7\times1079.8017-(62.17)^{2}]}}$
$r=\frac{7431.753-4473.197}{\sqrt{(8166.928 - 5181.1204)(7558.6119 - 3865.1089)}}$
$r=\frac{2958.556}{\sqrt{(2985.8076)(3693.503)}}$
$r=\frac{2958.556}{\sqrt{11028794.73}}$
$r=\frac{2958.556}{3320.962}$
$r\approx0.89$
$R^{2}=r^{2}=(0.89)^{2}=0.7921\approx0.79$

The coefficient of determination $R^{2}$ represents the proportion of the variance in the dependent variable (FFQ) that is explained by the independent variable (DR). So, 79% of the variability in $y$ (FFQ) is explained by the regression of $y$ on $x$ (DR). Also, the relationship between the two variables is linear (not non - linear as stated in option c).

Answer:

A. The coefficient of determination is 0.79
E. 79% of the variability in y is explained by the regression of y on x