Question

the manager at a new bank wants to hire enough tellers to ensure no customer waits in line too long. to gather more information, she waits in line at other local banks. at each bank the manager notes the number of teller windows that are open, x, as well as the number of minutes she has to wait in line before being served, y. number of teller windows open 2 2 3 4 6 time spent waiting in line (in minutes) 11 12 12 14 4 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, we need to calculate the sums: $\sum x$, $\sum y$, $\sum x^{2}$, $\sum y^{2}$, $\sum xy$. Let $x$ be the number of open teller - windows and $y$ be the time spent waiting in line.
We have $n = 5$ data points: $(x_1,y_1)=(2,11)$, $(x_2,y_2)=(2,12)$, $(x_3,y_3)=(3,12)$, $(x_4,y_4)=(4,14)$, $(x_5,y_5)=(6,4)$.

Step2: Calculate $\sum x$, $\sum y$, $\sum x^{2}$, $\sum y^{2}$, $\sum xy$

• $\sum x=2 + 2+3 + 4+6=17$
• $\sum y=11 + 12+12+14+4=53$
• $\sum x^{2}=2^{2}+2^{2}+3^{2}+4^{2}+6^{2}=4 + 4+9+16+36=69$
• $\sum y^{2}=11^{2}+12^{2}+12^{2}+14^{2}+4^{2}=121+144+144+196+16=621$
• $\sum xy=(2\times11)+(2\times12)+(3\times12)+(4\times14)+(6\times4)=22 + 24+36+56+24=162$

Step3: Substitute into the formula

$n = 5$.

• $n\sum xy=5\times162 = 810$
• $(\sum x)(\sum y)=17\times53 = 901$
• $n\sum x^{2}=5\times69 = 345$
• $(\sum x)^{2}=17^{2}=289$
• $n\sum y^{2}=5\times621 = 3105$
• $(\sum y)^{2}=53^{2}=2809$

The denominator:
\[

$$\begin{align*} &\sqrt{(n\sum x^{2}-(\sum x)^{2})(n\sum y^{2}-(\sum y)^{2})}\\ =&\sqrt{(345 - 289)(3105 - 2809)}\\ =&\sqrt{56\times296}\\ =&\sqrt{16576}\\ =&128.75 \end{align*}$$

\]

The numerator: $n(\sum xy)-(\sum x)(\sum y)=810 - 901=-91$

So, $r=\frac{-91}{128.75}\approx - 0.706$

Answer:

$-0.706$