Question

Does Elevation Affect Temperature Mid-June?
Suppose that you wanted to determine the effect, if any, that elevation has on temperature. The table below lists the elevations (in feet above sea level) of 24 randomly selected cities in the United States and the low temperatures (in degrees Fahrenheit) of these cities on June 15, 2020.

Elevation	1365	−282	5280	4551	6910	6063	3875	2730	7	1201	2001	1843
Low Temp.	56	79	56	56	39	55	42	51	74	63	73	48
Elevation	3202	2389	4226	1550	2134	2080	−7	141	909	50	338	1086
Low Temp.	64	73	56	53	58	57	80	55	74	56	69	78

Our eyes can be fooled by how strong a linear relationship is; we need to use a numerical measure, the correlation coefficient, to accurately describe the association between elevation and low temperature. Correlation measures the strength and direction (type) of linear relationships.

This is quite tedious to compute by hand, so we will use our calculator to obtain the value of r.

Using your calculator, press STAT, choose CALC, then 8: LinReg (a+bx). Enter L₁, L₂. Then press the ENTER key to get the correlation coefficient (rounded to 4 decimal places).

r ≈                                           

Given the value of r above, what is the strength of the linear relationship?            Weak        Moderate      Strong

When a scatterplot shows a linear relationship, we would like to summarize the overall pattern by drawing a line on the scatterplot. A regression line describes how a response variable changes as an explanatory variable changes. The least-squares regression line is the line that makes the sum of the squares of the vertical distances of the data points to the line as small as possible.

The form of the equation of a line is y = a + bx where a is the y-intercept, the value of y when x=0, and b is the slope, the amount y changes when x increases by one unit.

Using your calculator, press STAT, choose CALC, 8: LinReg (a+bx). Enter L₁, L₂. Then press the ENTER key to get the coefficients for the least-squares regression line (don't round).  

a =                                                      

b =                                                      

The equation of the least-squares regression line is: _________________________________________________

The coefficient of determination, r², is the proportion of the variation in the values of y that is explained by the linear relationship with x.

r² ≈                                 for this data (write as a decimal rounded to four decimal places)

Answer 1

r ≈ -0.6114

Given the value of r above, what is the strength of the linear relationship?            Moderate     

The form of the equation of a line is y = a + bx where a is the y-intercept, the value of y when x=0, and b is the slope, the amount y changes when x increases by one unit.

a = 68.872408335884

b =    -0.00350355691549934                         

The equation of the least-squares regression line is: y=68.872408335884-0.00350355691549934x

The coefficient of determination, r², is the proportion of the variation in the values of y that is explained by the linear relationship with x.

r² ≈ 0.3738 for this data (write as a decimal rounded to four decimal places)

Regression Analysis
	r²	0.3738
	r	-0.6114
	Std. Error	9.358
	n	24
	k	1
	.
ANOVA table
Source	SS	df	MS	F	p-value
Regression	1,150.1900	1	1,150.1900	13.13	.0015
Residual	1,926.7683	22	87.5804
Total	3,076.9583	23
Regression output				confidence interval
variables	coefficients	std. error	t (df=22)	p-value	95% lower
Intercept	68.8724
x	-0.0035	0.0010	-3.624	.0015	-0.0055

The given data is:

Elevation	Low Temp
1365	56
-282	79
5280	56
4551	56
6910	39
6063	55
3875	42
2730	51
7	74
1201	63
2001	73
1843	48
3202	64
2389	73
4226	56
1550	53
2134	58
2080	57
-7	80
141	55
909	74
50	56
338	69
1086	78