Answer to Question #201482 in Statistics and Probability for John

Question

AUS Rural Bank is studying the relationship between the mean account balance for individual accounts and the number of transactions per month. A sample of eight accounts revealed:

customer 1 2 3 4 5 6 7 8

Mean balance 50 32 53 24 15 10 16 27

No of transactions 5 2 7 9 10 2 4 11

Find the coefficient of correlation. Determine at the 0.01 significance level whether the correlation in the population is greater than zero.

Step 1: state the hypotheses,

Ho: __________________________________________________________

H1: __________________________________________________________

Step 2: The level of significance and critical region: α=__________

And t critical______________

Step 3: Complete the table and compute for the value of r and t

customer X Y X2 Y2 XY

1

2

3

4

5

6

7

8

∑▒〖X=〗

∑▒Y=

∑▒〖X^2=〗

∑▒〖Y^2=〗

∑▒〖XY=〗

r=

tcomputed=

Step 4: Decision rule:

Step 5: Conclusion

Answer 1

The coefficient of correlation between two variables can be obtained by:

$r=\frac{\sum (y-\bar{y}) \times (x-\bar{x})}{\sqrt{\sum (x-]bar{x})^2 \times \sum (y-\bar{y})^2}} \\ = \frac{13.25}{\sqrt{87.5 \times 1777.875}} \\ = 0.0336$

H₀: ρ=0

H₁: ρ>0

The level of significance and critical region: α=0.01

n=8

d.f.=n-1=7

$t= \frac{r\times \sqrt{n-2}}{\sqrt{1-r^2}} \\ = \frac{0.0336 \times \sqrt{8-2}}{\sqrt{1 -0.0336^2}} \\ = \frac{0.0822881}{0.999436} \\ = 0.082335$

The p-value can be obtained as:

The chart value of probability for t=0.082335 with d.f.=7 is 0.5315

So, p-value for a right tailed test is

p-value=1-0.5315 = 0.4685

Decision rule: We reject the H₀ if the p-value of the test is less than α=0.01.

P-value > 0.01

So, we do not have enough evidence against H₀ to reject it, so we fail to reject the H₀.

We can conclude, that the correlation in the population is not greater than zero.