Question

a fitness app tracked daily step counts. the results are shown in the table below. what is the correlation coefficient, to the nearest thousandth, and the strength of the linear model for this data?
day\tsteps walked
1\t30
2\t31
3\t21
4\t39
5\t19
options:
-0.272 and strong
-0.222 and strong
-0.272 and weak
-0.222 and weak

Explanation:

Step1: Identify variables

Let \( x \) be the day (\( 1, 2, 3, 4, 5 \)) and \( y \) be the steps walked (\( 30, 31, 21, 39, 19 \)).

Step2: Calculate necessary sums

• \( n = 5 \)
• \( \sum x = 1 + 2 + 3 + 4 + 5 = 15 \)
• \( \sum y = 30 + 31 + 21 + 39 + 19 = 140 \)
• \( \sum xy = (1)(30) + (2)(31) + (3)(21) + (4)(39) + (5)(19) = 30 + 62 + 63 + 156 + 95 = 406 \)
• \( \sum x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25 = 55 \)
• \( \sum y^2 = 30^2 + 31^2 + 21^2 + 39^2 + 19^2 = 900 + 961 + 441 + 1521 + 361 = 4184 \)

Step3: Use correlation coefficient formula

The formula for the correlation coefficient \( r \) is:
\[
r = \frac{n\sum xy - \sum x \sum y}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}
\]
Substitute the values:

• Numerator: \( 5(406) - 15(140) = 2030 - 2100 = -70 \)
• Denominator part 1: \( 5(55) - 15^2 = 275 - 225 = 50 \)
• Denominator part 2: \( 5(4184) - 140^2 = 20920 - 19600 = 1320 \)
• Denominator: \( \sqrt{50 \times 1320} = \sqrt{66000} \approx 256.9046 \)
• \( r = \frac{-70}{256.9046} \approx -0.272 \)

Step4: Determine strength

A correlation coefficient with magnitude close to 0 (between -0.3 and 0.3) indicates a weak linear relationship. Since \( |r| \approx 0.272 < 0.3 \), the linear model is weak.

Answer:

\(-0.272\) and weak (corresponding to the option: \(-0.272\) and weak)