Question 1175482: x y
0 0.99527
1 0.9907
2 0.98513
3 0.98256
4 0.97799
5 0.98042
6 0.98785
7 0.98428
8 0.97471
9 0.96514
10 0.97457
11 0.976
(a) Use the linear correlation coefficient to determine if there is correlation. r =
(b) According to the linear regression equation, the bone density of someone who drinks 20 colas per week is
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let's solve this problem step-by-step.
**(a) Linear Correlation Coefficient (r)**
To find the linear correlation coefficient (r), we'll use a calculator or statistical software. Here's how to do it manually, but it's much easier to use technology.
1. **Calculate Σx, Σy, Σx², Σy², and Σxy:**
* Σx = 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 66
* Σy = 0.99527 + 0.9907 + 0.98513 + 0.98256 + 0.97799 + 0.98042 + 0.98785 + 0.98428 + 0.97471 + 0.96514 + 0.97457 + 0.976 = 11.77462
* Σx² = 0² + 1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² + 10² + 11² = 506
* Σy² = 0.99527² + 0.9907² + 0.98513² + 0.98256² + 0.97799² + 0.98042² + 0.98785² + 0.98428² + 0.97471² + 0.96514² + 0.97457² + 0.976² = 11.5549060614
* Σxy = (0 * 0.99527) + (1 * 0.9907) + (2 * 0.98513) + (3 * 0.98256) + (4 * 0.97799) + (5 * 0.98042) + (6 * 0.98785) + (7 * 0.98428) + (8 * 0.97471) + (9 * 0.96514) + (10 * 0.97457) + (11 * 0.976) = 64.77098
2. **Apply the formula for r:**
* r = (nΣxy - (Σx)(Σy)) / √((nΣx² - (Σx)²)(nΣy² - (Σy)²))
* n = 12
3. **Plug in the values:**
* r = (12 * 64.77098 - (66 * 11.77462)) / √((12 * 506 - 66²)(12 * 11.5549060614 - 11.77462²))
* r = (777.25176 - 777.12492) / √((6072 - 4356)(138.6588727368 - 138.6416625844))
* r = 0.12684 / √((1716)(0.0172101524))
* r = 0.12684 / √(29.5326915264)
* r = 0.12684 / 5.4344
* r ≈ 0.02334
**Therefore, r ≈ 0.02334.**
**Interpretation:** Since r is very close to 0, there is very weak or no linear correlation between x and y.
**(b) Linear Regression Equation and Bone Density at x = 20**
1. **Find the Linear Regression Equation (y = mx + b):**
* **Slope (m):**
* m = (nΣxy - (Σx)(Σy)) / (nΣx² - (Σx)²)
* m = 0.12684 / 1716
* m ≈ 0.0000739
* **Y-intercept (b):**
* b = (Σy - mΣx) / n
* b = (11.77462 - (0.0000739 * 66)) / 12
* b = (11.77462 - 0.0048774) / 12
* b = 11.7697426 / 12
* b ≈ 0.9808118833
* **Linear Regression Equation:**
* y = 0.0000739x + 0.9808118833
2. **Find Bone Density at x = 20:**
* y = 0.0000739(20) + 0.9808118833
* y = 0.001478 + 0.9808118833
* y ≈ 0.9822898833
**Therefore, according to the linear regression equation, the bone density of someone who drinks 20 colas per week is approximately 0.9823.**
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