Question

Just need question 2a-e answered. Thanks!

The table gives Distance X from the lake in miles, and Snowfall Y in inches for seven weather stations. Case X-MX Y-My (X-Mx )(Y-My) (X-Mx)? (Y-My) 2 3 4 5 6 7 Sum X 104 102 92 96 101 78 69 Y 8 10 13 16 14 21 26 NA NA 1. Explore the data: create a scatterplot. 1a. Type the data into a blank SPSS spreadsheet. Name variables as Distance and Snowfall respectively. Go to Graphs-Legacy Dialogs-Scatter/Dot-Simple Scatter-Define. In the window that follows, select Distance into X axis and Snowfall into Y axis. Click on OK. 1b.Double click on the scatter plot to activate it. Double click on the horizontal axis and select the Scale tab. At Auto, uncheck all boxes. At Custom, set 0 for Minimum, 110 for Maximum, 20 for Major increments, and 0 for Origin. Click on Apply. Within the same window, activate the vertical axis and set O for Minimum, 70 for Maximum, 20 for Major increments, and 0 for Origin. Ic. Continue to go to Options-Reference Line from Equation. A line appears from the origin with a positive slope. The Custom Equation window contains an equation. Enter a negative slope to change the line's direction. An approximate fitting value is -0.47, but try a few different values first to see how they will alter the line's position. Do the same to the intercept (a right value is around 58). 1d. Go to Options-X Axis Reference Line. A vertical line appears. Double click the line so that the mouse curser turns into a cross on the line. Use the cross to grab the line, and move it to line up with 0 on the horizontal axis. Similarly, create a horizontal line to line up with 0 on the vertical axis. Use Option-Title to give the graph a title. Close Chart Editor. You will come back to revise this graph later.

1. Explore the data: create a scatterplot. 1a. Type the data into a blank SPSS spreadsheet. Name variables as Distance and Snowfall respectively. Go to Graphs-Legacy Dialogs-Scatter/Dot-Simple Scatter-Define. In the window that follows, select Distance into X axis and Snowfall into Y axis. Click on OK. 1b.Double click on the scatter plot to activate it. Double click on the horizontal axis and select the Scale tab. At Auto, uncheck all boxes. At Custom, set 0 for Minimum, 110 for Maximum, 20 for Major increments, and 0 for Origin. Click on Apply. Within the same window, activate the vertical axis and set 0 for Minimum, 70 for Maximum, 20 for Major increments, and 0 for Origin. 1c. Continue to go to Options-Reference Line from Equation. A line appears from the origin with a positive slope. The Custom Equation window contains an equation. Enter a negative slope to change the line’s direction. An approximate fitting value is -0.47, but try a few different values first to see how they will alter the line’s position. Do the same to the intercept (a right value is around 58). 1d. Go to Options-X Axis Reference Line. A vertical line appears. Double click the line so that the mouse curser turns into a cross on the line. Use the cross to grab the line, and move it to line up with 0 on the horizontal axis. Similarly, create a horizontal line to line up with 0 on the vertical axis. Use Option-Title to give the graph a title. Close Chart Editor. You will come back to revise this graph later.

2. Fitting the regression line: complete the following using the above table for your calculations

2a. Calculate means of the two variables MX = MY=

2b. Write down the formula and value of the covariance between the two variables 2

2c. Calculate the two variables’ standard deviations sX = sY=

2d. Write down the formula and value of the correlation coefficient between the two variables

2e. Write down the formula and value for the best estimate of slope b and intercept a respectively 2f. Based on 2e, write out the regression equation in which distance is the independent variable and snowfall the dependent variable. Use variable names, do not use letters X and Y.

Answer 1

x	y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
104	8	150.9388	55.1837	-91.265
102	10	105.7959	29.4694	-55.837
92	13	0.0816	5.8980	-0.694
96	16	18.3673	0.3265	2.449
101	14	86.2245	2.0408	-13.265
78	21	188.0816	31.0408	-76.408
69	26	515.9388	111.7551	-240.122

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	642.00	108.00	1065.43	235.71	-475.14
mean	91.71	15.43	SSxx	SSyy	SSxy

a)

Sample size,   n =   7
here, x̅ = Σx / n=   91.714  
ȳ = Σy/n =   15.429  
.

b)

covariance = Σ(x-x̅)(y-ȳ) /(n-1) =    -79.1905
.

c)

Sx =√ [ Σ(X - X̄)²/(n-1)] =13.32

Sy =√ [ Σ(y - ȳ)²/(n-1)] =6.267

...........

d)

correlation coefficient ,    r = SSxy/√(SSx.SSy) =   -0.94813

....

e)

estimated slope , ß1 = SSxy/SSxx =   -475.1429/1065.4286=   -0.4460
intercept,ß0 = y̅-ß1* x̄ =   15.4286- (-0.446 )*91.7143=   56.3298
...

f)

Regression line is,

snowfall=   56.33   + (   -0.446   )*distance

.............

Please let me know in case of any doubt.

Thanks in advance!

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