Question 13.7: Bank Branches and Deposits A researcher wishes to see if the......

Step 1 State the hypotheses.

H_{0}: \rho=0 and H_{1}: \rho \neq 0

Step 2 Find the critical value. Use Table L to find the value for n=8 and \alpha=0.05. It is \pm 0.738. See Figure 13-3.

Step 3 Find the test value.

a. Rank each data set as shown in the table.

\begin{array}{|c|cc|rc|} \hline \text{Bank }& \text{Branches }& \text{Rank }& \text{Deposits }& \text{Rank }\\ \hline A & 209 & 5 & 23 & 5 \\ B & 353 & 7 & 31 & 8 \\ C & 19 & 1 & 7 & 3 \\ D & 201 & 4 & 12 & 4 \\ E & 344 & 6 & 26 & 7 \\ F & 132 & 3 & 5 & 2 \\ G & 401 & 8 & 24 & 6 \\ H & 126 & 2 & 4 & 1 \\ \hline \end{array}

Let X_{1} be the rank of the branches and X_{2} be the rank of the deposits. b. Subtract the ranking \left(X_{1}-X_{2}\right).

5-5=0 \quad 7-8=-1 \quad 1-3=-2 \quad \text { etc. }

c. Square the differences.

0^{2}=0 \quad(-1)^{2}=1 \quad(-2)^{2}=4 \quad \text { etc. }

d. Find the sum of the squares.

0+1+4+0+1+1+4+1=12

The results can be summarized in a table as shown.

\begin{array}{|c|c|c|r|} \hline \boldsymbol{X}_{1} & \boldsymbol{X}_{\mathbf{2}} & \boldsymbol{d}=\boldsymbol{X}_{1}-\boldsymbol{X}_{\mathbf{2}} & \boldsymbol{d}^{\mathbf{2}} \\ \hline 5 & 5 & 0 & 0 \\ 7 & 8 & -1 & 1 \\ 1 & 3 & -2 & 4 \\ 4 & 4 & 0 & 0 \\ 6 & 7 & -1 & 1 \\ 3 & 2 & 1 & 1 \\ 8 & 6 & 2 & 4 \\ 2 & 1 & 1 & 1 \\ & & & \Sigma d^{2}=\overline{12} \\ \hline \end{array}

e. Substitute in the formula for r_{s}.

\begin{aligned}& r_{s}=1-\frac{6 \Sigma d^{2}}{n\left(n^{2}-1\right)} \quad \text { where } n=\text { number of pairs } \\& r_{s}=1-\frac{6 \cdot 12}{8\left(8^{2}-1\right)}=1-\frac{72}{504}=0.857\end{aligned}

Step 4 Make the decision. Reject the null hypothesis since r_{s}=0.857, which is greater than the critical value of 0.738.

Step 5 Summarize the results. There is enough evidence to say that there is a linear relationship between the number of branches a bank has and the deposits of the bank.