Question

a study of bone density on 5 random women at a hospital produced the following results. age 33 45 53 57 65 bone density 350 345 340 320 315 step 3 of 3 : calculate the correlation coefficient, r. round your answer to three decimal places.

Answer 1

The correlation coefficient (r) is 0.599, rounded to three decimal places.

How to find correlation coefficient?

To calculate the correlation coefficient (r), use the following formula:

`\[ r = (n(\sum xy) - (\sum x)(\sum y))/(√([n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2])) \]`

where:

n = number of data points,

x and y = variables (age and bone density, in this case),

∑xy = sum of the product of corresponding values of x and y,

∑x and ∑y = sums of x and y,

∑x² and ∑y² = sums of the squares of x and y.

Calculate the values:

`\[ \sum x = 33 + 45 + 53 + 57 + 65 = 253 \]`

`\[ \sum y = 350 + 345 + 340 + 320 + 315 = 1670 \]`

`\[ \sum xy = (33 * 350) + (45 * 345) + (53 * 340) + (57 * 320) + (65 * 315) = 857600 \]`

`\[ \sum x^2 = 33^2 + 45^2 + 53^2 + 57^2 + 65^2 = 9322 \]`

`\[ \sum y^2 = 350^2 + 345^2 + 340^2 + 320^2 + 315^2 = 1107550 \]`

Now, substitute these values into the correlation coefficient formula:

`\[ r = (5(\sum xy) - (\sum x)(\sum y))/(√([5\sum x^2 - (\sum x)^2][5\sum y^2 - (\sum y)^2])) \]`

`\[ r = (5(857600) - (253)(1670))/(√([5(9322) - (253)^2][5(1107550) - (1670)^2])) \]`

`\[ r = (4288000 - 422810)/(√([46610][5551930 - 2788900])) \]`

`\[ r = (3865190)/(√(46610 * 2763030)) \]`

`\[ r \approx (3865190)/(645448.413) \]`

r = 0.599

Therefore, the correlation coefficient (r) is 0.599, rounded to three decimal places.