Question

 

An environmentalist wants to determine the relationship between the number of fires, in thousands, and the number of acres burned, in hundreds of thousands. Based on this data, decide if the correlation is significant at alpha = 0.05.

Number of fires x	73	74	58	48	80	65	54	49
Number of acres burned y	64	46	22	23	51	12	29	10

1. Draw a scatter plot of the data.

2. Showing all work calculate r.

3. Showing all work determine where to reject or not reject the null hypothesis.

4. Determine the line of best fit y = a + bx.

5. When x = 61 what is y and what does it mean I this context.

6. Determine r^2.

7. Determine Sset.

8. Find the 95% prediction interval when x = 61.

Answer 1

1)

2. Showing all work calculate r.

	X	Y	X*Y	X2	Y2
	73	64	4672	5329	4096
	74	46	3404	5476	2116
	58	22	1276	3364	484
	48	23	1104	2304	529
	80	51	4080	6400	2601
	65	12	780	4225	144
	54	29	1566	2916	841
	49	10	490	2401	100
Sum =	501	257	17372	32415	10911

3. Showing all work determine where to reject or not reject the null hypothesis.

The hypothesis being tested is:

H₀: β₁ = 0

H₁: β₁ ≠ 0

ANOVA table
Source	SS	df	MS	F	p-value
Regression	1,569.1183	1	1,569.1183	8.67	.0258
Residual	1,085.7567	6	180.9595
Total	2,654.8750	7

Since p value

There is sufficient evidence to conclude that β₁ not equal to zero at 5% level of significance.

4. Determine the line of best fit y = a + bx.

5. When x = 61 what is y and what does it mean I this context.

This means that when number of fires is 61 then the predicted value of the number of acres burned is 30.1293 n hundreds of thousands.

6. Determine r^2.

r=0.769

r^2=0.769^2=0.5914

7. Determine Sset=13.452

Using excel

Regression Analysis
	r²	0.591
	r	0.769
	Std. Error	13.452
	n	8
	k	1
	Dep. Var.	Number of acres burned,y
ANOVA table
Source	SS	df	MS	F	p-value
Regression	1,569.1183	1	1,569.1183	8.67	.0258
Residual	1,085.7567	6	180.9595
Total	2,654.8750	7
Regression output				confidence interval
variables	coefficients	std. error	t (df=6)	p-value	95% lower	95% upper
Intercept	-44.8031	26.55	-1.69	0.14	-109.78	20.17
Number of fires,x	1.2284	0.4172	2.945	.0258	0.2076	2.2491
Predicted values for: Number of acres burned,y
		95% Confidence Interval	95% Prediction Interval
Number of fires,x	Predicted	lower	upper	lower	upper	Leverage
61	30.129	18.374	41.884	-4.823	65.081	0.128

8. Find the 95% prediction interval when x = 61.

The 95% prediction interval is (-4.823,65.081)