claudia records the hours she spent studying and her test scores for 5 tests. what is the correlation…

Answer:

1. Correlation coefficient: First, we need to use the formula for the correlation coefficient $r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}$.
   • Let $x$ be the hours - spent studying and $y$ be the test score.
   • Calculate the necessary sums:
     • $n = 5$.
     • $\sum x=1 + 2+3 + 4+5=15$.
     • $\sum y=72 + 80+90+82+95 = 419$.
     • $\sum xy=1\times72+2\times80 + 3\times90+4\times82+5\times95=72 + 160+270+328+475 = 1305$.
     • $\sum x^{2}=1^{2}+2^{2}+3^{2}+4^{2}+5^{2}=1 + 4+9+16+25 = 55$.
     • $\sum y^{2}=72^{2}+80^{2}+90^{2}+82^{2}+95^{2}=5184+6400+8100+6724+9025 = 35433$.
   • Substitute into the formula:
     • Numerator: $n(\sum xy)-(\sum x)(\sum y)=5\times1305-15\times419 = 6525-6285 = 240$.
     • Denominator - part 1: $n\sum x^{2}-(\sum x)^{2}=5\times55 - 15^{2}=275 - 225 = 50$.
     • Denominator - part 2: $n\sum y^{2}-(\sum y)^{2}=5\times35433-419^{2}=177165 - 175561=1604$.
     • Denominator: $\sqrt{50\times1604}=\sqrt{80200}\approx283.196$.
     • $r=\frac{240}{283.196}\approx0.85$.
2. Strength of the model: Since the correlation coefficient $r\approx0.85$, and $|r|$ is close to 1 (where $|r|\in[0,1]$), the strength of the model is strong.

Explanation:

Step1: Calculate sums

Calculated $\sum x,\sum y,\sum xy,\sum x^{2},\sum y^{2}$ and $n$.

Step2: Calculate numerator

Found $n(\sum xy)-(\sum x)(\sum y)$.

Step3: Calculate denominator - parts

Found $n\sum x^{2}-(\sum x)^{2}$ and $n\sum y^{2}-(\sum y)^{2}$.

Step4: Calculate denominator

Took square - root of product of denominator parts.

Step5: Calculate correlation coefficient

Divided numerator by denominator.

Step6: Determine strength of model

Assessed strength based on value of $|r|$.