Question

pamela is a programmer in silicon valley and spends most of her day typing and drinking coffee. she wants to know if drinking coffee makes her type faster. over the next few days, pamela varies the number of cups of coffee she drinks, x, and measures the average number of words typed per minute on each day, y.
cups of coffee words typed
0.6 91
2.1 92
2.7 94
3.1 96
3.4 98
round your answer to the nearest thousandth.

Explanation:

Step1: Recall correlation - coefficient formula

The formula for the correlation coefficient $r$ is $r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}$. First, calculate the necessary sums:
Let $x$ be the number of cups of coffee and $y$ be the number of words typed.
$n = 5$
$\sum x=0.6 + 2.1+2.7 + 3.1+3.4=11.9$
$\sum y=91 + 92+94+96+98 = 471$
$\sum xy=(0.6\times91)+(2.1\times92)+(2.7\times94)+(3.1\times96)+(3.4\times98)$
$=54.6+193.2 + 253.8+297.6+333.2=1132.4$
$\sum x^{2}=0.6^{2}+2.1^{2}+2.7^{2}+3.1^{2}+3.4^{2}=0.36 + 4.41+7.29+9.61+11.56=33.23$
$\sum y^{2}=91^{2}+92^{2}+94^{2}+96^{2}+98^{2}=8281+8464+8836+9216+9604 = 44401$

Step2: Calculate the numerator

$n(\sum xy)-(\sum x)(\sum y)=5\times1132.4-11.9\times471$
$=5662-5604.9 = 57.1$

Step3: Calculate the first - part of the denominator

$n\sum x^{2}-(\sum x)^{2}=5\times33.23-11.9^{2}=166.15 - 141.61=24.54$

Step4: Calculate the second - part of the denominator

$n\sum y^{2}-(\sum y)^{2}=5\times44401-471^{2}=222005-221841 = 164$

Step5: Calculate the denominator

$\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}=\sqrt{24.54\times164}=\sqrt{4024.56}\approx63.44$

Step6: Calculate the correlation coefficient

$r=\frac{57.1}{63.44}\approx0.9$

Answer:

$r\approx0.9$