We look at cholesterol levels versus systolic blood pressure and obtain the following points (173, 155), (166, 141), (144, 116), (137, 117), (196, 187), and (183, 160). Find the sample coefficient of correlation.
The Correct Answer and Explanation is:
To find the sample coefficient of correlation (r) between cholesterol levels (x) and systolic blood pressure (y), we use the Pearson correlation formula:r=n∑xy−∑x∑y[n∑x2−(∑x)2][n∑y2−(∑y)2]r = \frac{n\sum xy – \sum x \sum y}{\sqrt{[n\sum x^2 – (\sum x)^2][n\sum y^2 – (\sum y)^2]}}r=[n∑x2−(∑x)2][n∑y2−(∑y)2]n∑xy−∑x∑y
Given Data Points:
- (173, 155)
- (166, 141)
- (144, 116)
- (137, 117)
- (196, 187)
- (183, 160)
Step 1: Organize the data
| x (cholesterol) | y (systolic BP) | x² | y² | xy |
|---|---|---|---|---|
| 173 | 155 | 29929 | 24025 | 26815 |
| 166 | 141 | 27556 | 19881 | 23406 |
| 144 | 116 | 20736 | 13456 | 16704 |
| 137 | 117 | 18769 | 13689 | 16029 |
| 196 | 187 | 38416 | 34969 | 36652 |
| 183 | 160 | 33489 | 25600 | 29280 |
Step 2: Compute the sums
- ∑x=999\sum x = 999∑x=999
- ∑y=876\sum y = 876∑y=876
- ∑x2=168895\sum x^2 = 168895∑x2=168895
- ∑y2=131620\sum y^2 = 131620∑y2=131620
- ∑xy=148886\sum xy = 148886∑xy=148886
- n=6n = 6n=6
Step 3: Plug into the formula
r=6(148886)−(999)(876)[6(168895)−(999)2][6(131620)−(876)2]r = \frac{6(148886) – (999)(876)}{\sqrt{[6(168895) – (999)^2][6(131620) – (876)^2]}}r=[6(168895)−(999)2][6(131620)−(876)2]6(148886)−(999)(876)r=893316−875124[1013370−998001][789720−767376]r = \frac{893316 – 875124}{\sqrt{[1013370 – 998001][789720 – 767376]}}r=[1013370−998001][789720−767376]893316−875124r=18192154369⋅22344=181923451667136≈181925876.78≈0.3096r = \frac{18192}{\sqrt{154369 \cdot 22344}} = \frac{18192}{\sqrt{3451667136}} \approx \frac{18192}{5876.78} \approx 0.3096r=154369⋅2234418192=345166713618192≈5876.7818192≈0.3096
Final Answer:
r≈0.310\boxed{r \approx 0.310}r≈0.310
Explanation
The sample coefficient of correlation, commonly referred to as Pearson’s r, quantifies the strength and direction of the linear relationship between two quantitative variables. In this case, we analyze the relationship between cholesterol levels and systolic blood pressure using a sample of six data points.
Pearson’s r ranges from -1 to 1. An r value close to 1 indicates a strong positive linear correlation, meaning that as one variable increases, the other also increases. An r value close to -1 indicates a strong negative linear correlation, meaning that as one variable increases, the other decreases. An r value close to 0 suggests no linear correlation.
Using the formula, we calculated r ≈ 0.310. This result indicates a weak positive correlation between cholesterol level and systolic blood pressure in this small sample. In practical terms, this means that, on average, individuals with higher cholesterol levels also tend to have slightly higher systolic blood pressure, though the relationship is not particularly strong.
The relatively low value of r could be due to natural biological variation, measurement error, or the presence of other influencing factors not captured in this dataset, such as age, weight, or medical history. Also, with only six data points, any conclusions drawn must be interpreted cautiously, as small samples are more prone to outliers and noise affecting correlation.
Overall, this analysis provides a preliminary indication of a positive relationship between the two variables, but further study with a larger sample size would be needed to draw more robust conclusions.
