Chapter 19.4, Problem 99E

To determine

Calculate the p value.

Explanation of Solution

Null hypothesis (H₀) is ${\text{ρ}}_{\text{S}}=\text{0}$ and alternate hypothesis is (H₁) ${\text{ρ}}_{\text{S}}\ne \text{0}$.

The calculated correlation coefficient (Table ${\text{r}}_{\text{s}}$) is 0.409.

Table 1 shows the sum of different values.

Table 1

Experience	a	Rating	b	${\text{a}}^2$	${\text{b}}^2$	$\text{a}\times \text{b}$
1	1.5	1	2	2.25	4	3
17	13	4	16	169	256	208
20	17	4	16	289	256	272
9	6.5	55	21.5	42.25	462.25	139.75
2	3	2	6	9	36	18
13	11	4	16	121	256	176
9	6.5	3	11	42.25	121	71.5
23	23	5	21.5	529	462.25	494.5
7	4	2	6	16	36	24
10	8	5	21.5	64	462.25	172
12	10	5	21.5	100	462.25	215
24	24	2	6	576	36	144
8	5	2	6	25	36	30
20	17	5	21.5	289	462.25	365.5
21	20	3	11	400	121	220
19	15	2	6	225	36	90
1	1.5	1	2	2.25	4	3
22	22	3	11	484	121	242
20	17	4	16	289	256	272
11	9	3	11	81	121	99
18	14	5	21.5	196	462.25	301
14	12	4	16	144	256	192
21	20	3	11	400	121	220
21	20	1	2	400	4	40
Total	300		300	4,895	4,850.5	4,012.25

Standard deviation $\left({\text{s}}_{\text{ab}}\right)$ can be calculated as follows.

$\begin{array}{c}{\text{s}}_{\text{ab}}=\frac{\text{1}}{\text{n}-\text{1}}\left[{\displaystyle \sum _{\text{i}=\text{1}}^{\text{n}}{\text{a}}_{\text{i}}{\text{b}}_{\text{i}}}\text{-}\frac{{\displaystyle \sum _{\text{i}=\text{1}}^{\text{n}}{\text{a}}_{\text{i}}}{\displaystyle \sum _{\text{i}=\text{1}}^{\text{n}}{\text{b}}_{\text{i}}}}{\text{n}}\right]\\ =\frac{1}{24-1}\left[4,012.25-\frac{(300)(300)}{24}\right]\\ =11.40\end{array}$

Standard deviation is 11.4.

Standard deviation of ‘a’ $\left({\text{s}}_{\text{a}}\right)$ can be calculated as follows.

$\begin{array}{c}{\text{s}}_{\text{a}}=\sqrt{\frac{\text{1}}{\text{n}-\text{1}}\left[{\displaystyle \sum _{\text{i}=\text{1}}^{\text{n}}{\text{a}}_{\text{i}}^{\text{2}}}\text{-}\frac{{\left({\displaystyle \sum _{\text{i}=\text{1}}^{\text{n}}{\text{a}}_{\text{i}}}\right)}^{\text{2}}}{\text{n}}\right]}\\ =\frac{1}{24-1}\left[4,895-\frac{{(300)}^2}{24}\right]\\ =7.056\end{array}$

 Standard deviation is 7.056.

Standard deviation of ‘b’ can be calculated as follows.

$\begin{array}{c}{\text{s}}_{\text{b}}=\sqrt{\frac{\text{1}}{\text{n}-\text{1}}\left[{\displaystyle \sum _{\text{i}=\text{1}}^{\text{n}}{\text{b}}_{\text{i}}^{\text{2}}}\text{-}\frac{{\left({\displaystyle \sum _{\text{i}=\text{1}}^{\text{n}}{\text{b}}_{\text{i}}}\right)}^{\text{2}}}{\text{n}}\right]}\\ =\sqrt{\frac{1}{24-1}\left[4,850.5-\frac{{(300)}^2}{24}\right]}\\ =6.917\end{array}$

Standard deviation is6.917.

The correlation coefficient $\left({\text{r}}_{\text{s}}\right)$ can be calculated as follows.

$\begin{array}{c}{\text{r}}_{\text{S}}=\frac{{\text{s}}_{\text{ab}}}{{\text{s}}_{\text{b}}{\text{s}}_{\text{b}}}\\ =\frac{11.40}{(7.056)(6.917)}\\ =0.2336\end{array}$

The correlation coefficient is 0.2336. Since the calculated correlation coefficient is less than the table value, the alternate hypothesis is not accepted.