Question

Assignment #8                                                                       

Problem A: Linear Regression

1) Below is some data from the late 1800s and early 1900s indicating a correlation between the number of ministers in New England and the amount of Cuban rum imported to Boston during those years.

Year	Ministers	Cuban rum
1860	8376	63
1865	6406	48
1870	7005	53
1875	8486	64
1880	9595	72
1885	10643	80
1890	11265	85
1895	10071	76
1900	10547	80
1905	11008	83
1910	13885	105
1915	18559	140
1920	23024	175
1925	24185	183
1930	25434	192
1935	29238	221
1940	34705	262

1. Find the correlation coefficient between number of minister and amount of Cuban rum. Describe what it means.

2. Find a linear model that relates the number of ministers and the amount of rum. What are

the values of ?₀ and ?₁?

3. Interpret the slope of the least-squares regression line in context.

4. Use the linear model to predict the amount of rum barrels if the number of ministers was 8000? Is this an example of extrapolation or interpolation.

5. Find the residual for 23024 ministers.

6. Test whether a linear relation exists between the number of ministers and Cuban rum at the 5% level of significance. Show all steps including conditions.

Answer 1

Soln

	Ministers (X)	Cuban rum (Y)	X2	Y2	XY
	8376	63	70157376	3969	527688
	6406	48	41036836	2304	307488
	7005	53	49070025	2809	371265
	8486	64	72012196	4096	543104
	9595	72	92064025	5184	690840
	10643	80	1.13E+08	6400	851440
	11265	85	1.27E+08	7225	957525
	10071	76	1.01E+08	5776	765396
	10547	80	1.11E+08	6400	843760
	11008	83	1.21E+08	6889	913664
	13885	105	1.93E+08	11025	1457925
	18559	140	3.44E+08	19600	2598260
	23024	175	5.3E+08	30625	4029200
	24185	183	5.85E+08	33489	4425855
	25434	192	6.47E+08	36864	4883328
	29238	221	8.55E+08	48841	6461598
	34705	262	1.2E+09	68644	9092710
Total	262432	1982	5.26E+09	300140	39721046

Using the above formula and values, we get

Correlation Coefficient (r) = 0.99999

The positive sign of Correlation Coefficient indicates direct relationship between Ministers and Cuban rum ie as one variable increases, another also increases and vice versa. Also, the magnitude of r indicates very strong relationship between the two variables

2)

Let the regression equation be: Y = a + bX

Where

Slope(b) = {n*∑XY - ∑X *∑Y}/{n*∑X2 – (∑X)2 } = 0.5

and a = ∑Y/n – b*∑X/n = 4

Hence,

Cuban Rum = 0.01 * Ministers – 0.25

Where

β0 = -0.25

β1 = 0.01

3)

Interpretation of Slope

For every 1 unit increase in Ministers, Cuban Rum increases by 0.01 units

4)

When Ministers = 8000

Using the regression equation, we get:

Cuban Rum = 0.01 * 8000 – 0.25 = 60.30

This is an example of extrapolation

5)

From the above scatterplot we can conclude that there is a linear relationship between the two variables

Correlation Hypothesis Test

Alpha = 0.05

df = n-2 = 17-2 = 15

Null and Alternate Hypothesis

H0: ρ = 0 (No Correlation)

Ha: ρ <> 0 (Correlation is present)

n = 17

we will be doing a two-tailed hypothesis test

Test Statistic

t = r* (n-2)1/2/(1-r2)1/2 = 0.99999 *(17-2)1/2 /(1-0.999992)1/2 = 742.16

p-value = TDIST(742.16,17-2,2) = 1.16018E-35

Result

Since the p-value is less than 0.05, we reject the null hypothesis.

Conclusion

We conclude that there is correlation between Ministers and Cuban Rum