Question

wine intake x (grams of alcohol per day) 2.5 8.5 15.5 26.5 relative risk y 1.00 1.08 1.15 1.22 a. make a scatterplot of these data. based on the scatterplot, do you expect the correlation to be positive or negative? near ±1 or not? b. find the correlation r between wine intake and relative risk. do the data show that women who consume more wine tend to have higher relative risks of breast - cancer? c. can you conclude a causal relationship between wine consumption and breast cancer in women? explain.

Explanation:

Step1: Analyze scatter - plot expectation

As the value of wine intake (x) increases, the relative risk (y) also seems to increase. So we expect a positive correlation.

Step2: Calculate correlation coefficient formula

The formula for the correlation coefficient \(r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}\)
First, calculate the means: \(\bar{x}=\frac{2.5 + 8.5+15.5+26.5}{4}=\frac{53}{4}=13.25\), \(\bar{y}=\frac{1.00 + 1.08+1.15+1.22}{4}=\frac{4.45}{4}=1.1125\)
\(\sum_{i = 1}^{4}(x_{i}-\bar{x})(y_{i}-\bar{y})=(2.5 - 13.25)(1.00 - 1.1125)+(8.5 - 13.25)(1.08 - 1.1125)+(15.5 - 13.25)(1.15 - 1.1125)+(26.5 - 13.25)(1.22 - 1.1125)\)
\(=(- 10.75)\times(-0.1125)+(-4.75)\times(-0.0325)+(2.25)\times(0.0375)+(13.25)\times(0.1075)\)
\(=1.209375 + 0.154375+0.084375 + 1.424375=2.8725\)
\(\sum_{i = 1}^{4}(x_{i}-\bar{x})^{2}=(2.5 - 13.25)^{2}+(8.5 - 13.25)^{2}+(15.5 - 13.25)^{2}+(26.5 - 13.25)^{2}\)
\(=(-10.75)^{2}+(-4.75)^{2}+(2.25)^{2}+(13.25)^{2}\)
\(=115.5625+22.5625 + 5.0625+175.5625=318.75\)
\(\sum_{i = 1}^{4}(y_{i}-\bar{y})^{2}=(1.00 - 1.1125)^{2}+(1.08 - 1.1125)^{2}+(1.15 - 1.1125)^{2}+(1.22 - 1.1125)^{2}\)
\(=(-0.1125)^{2}+(-0.0325)^{2}+(0.0375)^{2}+(0.1075)^{2}\)
\(=0.01265625+0.00105625+0.00140625+0.01155625=0.026675\)
\(r=\frac{2.8725}{\sqrt{318.75\times0.026675}}=\frac{2.8725}{\sqrt{8.49453125}}\approx\frac{2.8725}{2.9145}\approx0.986\)
The value of \(r\) is close to \(+ 1\).

Step3: Analyze causal relationship

Just because there is a strong positive correlation does not mean there is a causal relationship. There could be other confounding variables (e.g., lifestyle factors, genetic factors) that affect both wine - intake and the risk of breast cancer.

Answer:

a. We expect a positive correlation based on the scatter - plot as the relative risk seems to increase with wine intake.
b. The correlation \(r\approx0.986\), which is positive and close to \(+1\). The data show that women who consume more wine tend to have higher relative risks of breast cancer.
c. We cannot conclude a causal relationship between wine consumption and breast cancer in women because correlation does not imply causation and there may be confounding variables.