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Quiz 3: Mean, Variance, Correlation and Regression

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Question 1

A set of values, xi, has ∑6 xi = 40 i=1 and ∑6 2 x i = 1600 i=1 . What are the sample mean, x, and sample variance, s2 ?

a)
x = 6.6˙6, s2 = 0
   b)
x = 6.66˙, s2 = 1333.˙3
c)
x = 6.66˙, s2 = 266.6˙6
   d)
x = 266.6˙6, s2 = 16.32

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
1∑6 40 x- = - xi = -- = 6.6˙6 6 i=1 ⌊ 6 ⌋ ( ∑6 ) ( ∑6 )2 s2 = --1--⌈ x2i - 1- xi ⌉ n- 1 i=1 n i=1 1 1 2 = 5(1600- 6(40) ) = 266.6˙6
Not correct. Choice (d) is false.

Question 2

Suppose 7 ∑ xi = 21 i=1 and 9 ∑ xi = 24. i=1 Find x8 + x9.

a)
3
   b)
-3
c)
45
   d)
There is not enough information provided.

 

Your answer is correct.
∑9 ∑7 x8 + x9 = xi - xi = 24 - 21 = 3. i=1 i=1
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 3

Consider the following frequency table (i = 1,…,4)
 

| xi |1 2 3 4 fi 3 6 2 1
Find the mean of this data set.
a)
6.25
   b)
6
c)
2
   d)
2.08˙3

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
x = 1- 12 ∑ i=14fixi.
x = -1 12(3 + 12 + 6 + 4) = 25 12 = 2.083.

Question 4

The following table gives the relation between pairs of data values (xi,yi) for i = 1,…,5.
 

| xi |1 2 3 4 5 yi 2 4 6 8 10
All the points lie on the line y = bx with slope b and correlation coefficient r. Find b and r.
a)
r = 2, b = 1
   b)
r = 2, b = 2
c)
r = 1, b = 2
   d)
r = 1, b = 1 2

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
The line is clearly y = 2x, so b = 2. Since the relationship is linear and the slope is positive, r = 1.
Not correct. Choice (d) is false.

Question 5

The correlation coefficient r satisfies

0 ≤ r2 ≤ 1.
Which of the following statements is true ?
a)
-1 ≤ r ≤ 1
   b)
r ≤-1
c)
r ≥ 1
   d)
r ≥ 1 or r ≤-1

 

Your answer is correct.
In general if x2 < a then -√a- < x < √a-.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 6

Calculate the correlation coefficient, r, (to 2dp) for the bivariate data

(xi,yi) : (0,0) (3,1.4) (6,2.6) (7,3.8) (9,7.2).
You may use the totals: ∑ ixi2 = 175 and ∑ iyi2 = 75.
a)
0.93
   b)
0.09
c)
0.97
   d)
None of these.

 

Your answer is correct.
∑ ixi = 25,∑ iyi = 15,∑ ixiyi = 111.2 so Sxx = 50,Syy = 30 and Sxy = 36.2 and r = -36.2-- √50×30 = 0.93 to 2dp.
Not correct. Choice (b) is false.
Check your calculations for Sxx, Syy and Sxy.
Not correct. Choice (c) is false.
Check your calculations for Sxx, Syy and Sxy.
Not correct. Choice (d) is false.
Check your calculations for Sxx, Syy and Sxy.

Question 7

The correlation coefficient for a set of bivariate data (xi,yi) is r = 0.87, where the xi are measured in inches and the yi are measured in lbs. A second analyst records the xi values in cm. (1 inch ≈ 2.5 cm). What is the second analyst’s value of the correlation coefficient (to 2dp)?

a)
0.35
   b)
0.87
c)
2.18
   d)
Unable to determine without knowing the yi units.

 

Not correct. Choice (a) is false.
How do you adjust r for a change of units?
Your answer is correct.
r is not affected by a change of units.
Not correct. Choice (c) is false.
r cannot exceed 1.
Not correct. Choice (d) is false.
How do you adjust r for a change of units?

Question 8

Find the correlation coefficient for 6 pairs of observations if the LSR line is y = 0.5 - 0.05x and if 81% of the variation in y is explained by regression on x.

a)
0.9
   b)
0.81
c)
-0.05
   d)
None of these.

 

Not correct. Choice (a) is false.
Check the slope of the LSR line.
Not correct. Choice (b) is false.
This is r2.
Not correct. Choice (c) is false.
Try again.
Your answer is correct.
The slope is negative and r2 = 0.81, so r = -0.9.

Question 9

For the bivariate data (x1,y1) (x2,y2) ⋅⋅⋅(xn,yn), the least squares regression line is fitted. The line is y = 2.51 - 4.1x. You know that the first data point is (x1,y1) = (0.1,2.0), so the residual at this point is:

a)
2.1
   b)
-0.1
c)
0.1
   d)
2.0

 

Not correct. Choice (a) is false.
You have found the y value on the LSR line at x = 0.1.
Your answer is correct.
At x = 0.1, the value of y on the LSR line is ŷ = 2.51 - 0.41 = 2.1. Therefore the residual at x = 0.1 is 2.0 - 2.1 = -0.1.
Not correct. Choice (c) is false.
Try again.
Not correct. Choice (d) is false.
Try again.

Question 10

A correlation coefficient of r = 0.8 is reported for a sample of pairs (xi,yi). Without any further information, this implies that:

a)
as the x values decrease, the y values increase.
   b)
80% of the variation in y is due to regression on x.
c)
the (xi,yi) are scattered about a straight line of unknown positive slope.
   d)
the (xi,yi) are scattered about a straight line of slope 0.8.

 

Not correct. Choice (a) is false.
This would be the answer for a negative value of r.
Not correct. Choice (b) is false.
This is the interpretation of r2 not of r.
Your answer is correct.
Not correct. Choice (d) is false.
r is not the slope of the LSR line.
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