Question

some studies have suggested that a nightly glass of wine may not only take the edge off a day but also improve health. is wine good for your health? a study of nearly 1.3 million middle - aged british women examined wine consumption and the risk of breast cancer. the researchers were interested in how risk changed as wine consumption increased. risk is based on breast cancer rates in drinkers relative to breast cancer rates in nondrinkers in the study, with higher values indicating greater risk. in particular, a value greater than 1 indicates a greater breast cancer rate than that of nondrinkers. wine intake is the mean wine intake, in grams of alcohol per day (where one glass of wine is approximately 10 grams of alcohol), of groups of women in the study who drank approximately the same amount of wine per week. the given table contains data for drinkers only.
wine intake
x relative (gramse risk y per da y)
2.5 1.00
8.5 1.08
15.5 1.15
26.5 1.22
do the data show that women who consume more wine tend to have higher relative risks of breast cancer?
o no
o yes
(b) find the correlation r between wine intake and relative risk. give your answer to three decimal places.
r =

Explanation:

Step1: Recall correlation formula

The formula for the correlation coefficient $r$ is $r=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{\sqrt{[n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}][n\sum_{i = 1}^{n}y_i^{2}-(\sum_{i = 1}^{n}y_i)^{2}]}}$. Let $x$ be the wine - intake (in grams per day) and $y$ be the relative risk. Assume we have $n = 4$ data points: $(x_1,y_1)=(2.5,1.00),(x_2,y_2)=(8.5,1.08),(x_3,y_3)=(15.5,1.15),(x_4,y_4)=(26.5,1.22)$.

Step2: Calculate sums

First, calculate $\sum_{i = 1}^{4}x_i=2.5 + 8.5+15.5 + 26.5=53$, $\sum_{i = 1}^{4}y_i=1.00 + 1.08+1.15 + 1.22=4.45$, $\sum_{i = 1}^{4}x_i^{2}=2.5^{2}+8.5^{2}+15.5^{2}+26.5^{2}=6.25 + 72.25+240.25+702.25 = 1021$, $\sum_{i = 1}^{4}y_i^{2}=1.00^{2}+1.08^{2}+1.15^{2}+1.22^{2}=1+1.1664 + 1.3225+1.4884 = 4.9773$, and $\sum_{i = 1}^{4}x_iy_i=2.5\times1.00+8.5\times1.08+15.5\times1.15+26.5\times1.22=2.5+9.18+17.825+32.33=61.835$.

Step3: Substitute into formula

$n = 4$.
$n\sum_{i = 1}^{n}x_iy_i=4\times61.835 = 247.34$, $\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i=53\times4.45 = 235.85$, $n\sum_{i = 1}^{n}x_i^{2}=4\times1021 = 4084$, $(\sum_{i = 1}^{n}x_i)^{2}=53^{2}=2809$, $n\sum_{i = 1}^{n}y_i^{2}=4\times4.9773 = 19.9092$, $(\sum_{i = 1}^{n}y_i)^{2}=4.45^{2}=19.8025$.
The denominator is $\sqrt{(4084 - 2809)(19.9092 - 19.8025)}=\sqrt{1275\times0.1067}=\sqrt{135.0425}\approx11.62$.
The numerator is $247.34−235.85 = 11.49$.
So $r=\frac{11.49}{11.62}\approx0.99$.

Answer:

(a) Yes
(b) $r\approx0.99$