# Variable X Variable Y 1 205 1500 2 70 750 3 199 1500 4 151...

#	Variable X	Variable Y
1	205	1500
2	70	750
3	199	1500
4	151	1250
5	181	1250
6	217	1250
7	94	1000
8	298	2000
9	135	1000
10	211	1500
11	116	1250
12	72	500
13	82	500
14	206	1500
15	245	2000
16	219	1500
17	63	750
18	200	1500
19	151	1250
20	44	500

What is the significance of the slope. Use a significance level of 0.05

What is the coefficient of determination

Expert Solution

X	Y	*XY**	X2	Y2
205.00	1,500.00	3,07,500.00	42,025.00	22,50,000.00
70.00	750.00	52,500.00	4,900.00	5,62,500.00
199.00	1,500.00	2,98,500.00	39,601.00	22,50,000.00
151.00	1,250.00	1,88,750.00	22,801.00	15,62,500.00
181.00	1,250.00	2,26,250.00	32,761.00	15,62,500.00
217.00	1,250.00	2,71,250.00	47,089.00	15,62,500.00
94.00	1,000.00	94,000.00	8,836.00	10,00,000.00
298.00	2,000.00	5,96,000.00	88,804.00	40,00,000.00
135.00	1,000.00	1,35,000.00	18,225.00	10,00,000.00
211.00	1,500.00	3,16,500.00	44,521.00	22,50,000.00
116.00	1,250.00	1,45,000.00	13,456.00	15,62,500.00
72.00	500.00	36,000.00	5,184.00	2,50,000.00
82.00	500.00	41,000.00	6,724.00	2,50,000.00
206.00	1,500.00	3,09,000.00	42,436.00	22,50,000.00
245.00	2,000.00	4,90,000.00	60,025.00	40,00,000.00
219.00	1,500.00	3,28,500.00	47,961.00	22,50,000.00
63.00	750.00	47,250.00	3,969.00	5,62,500.00
200.00	1,500.00	3,00,000.00	40,000.00	22,50,000.00
151.00	1,250.00	1,88,750.00	22,801.00	15,62,500.00
44.00	500.00	22,000.00	1,936.00	2,50,000.00
3,159.00	24,250.00	43,93,750.00	5,94,055.00	3,31,87,500.00

CORRELATION

Based on the above table, the following is calculated:

Therefore, based on this information, the sample correlation coefficient is computed as follows:

Correlation Significance Check

Let ρ = population correlation coefficient (unknown)
r = sample correlation coefficient (known; calculated from sample data)

Null and Alternative hypothesis:
H0:ρ=0 i.e. population correlation coefficient is 'close to 0'
HA:ρ≠0 i.e. population correlation coefficient is 'significantly different from 0'

Value of correlation coefficient:
r=0.9393

Degrees of freedom
The sample size is n=20, so then the number of degrees of freedom is df = n-2 = 20 - 2 = 18

Table of Critical Values Method
Critical region:
The corresponding critical correlation value rc for a significance level of α=0.05, for a Two-tailed test is: rc=0.4438

Rejection Region
Observe that in this case, the null hypothesis H0:ρ=0 is rejected if |r|>rc=0.4438.

Decision about the null hypothesis
Based on the sample correlation provided, we have that |r|=0.9393>rc=0.4438, from which is concluded that the null hypothesis is rejected.

T-statistic Method
Test statistic:
The value of the test statistic is calculated using the following formula:

Decision about the null hypothesis:
Since it is observed that |t|=11.6138 > tc=2.1009, it is then concluded that the null hypothesis is rejected.

The p-value is 0
Using the P-value approach: The p-value is p=0, and since p=0≤0.05, it is concluded that the null hypothesis is rejected.

Conclusion
It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the correlation coefficient is different than 0, at the 0.05 significance level.

Coefficient of determination

orchestra answered 2 years ago

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