Applied Statistics: From Bivariate Through Multivariate Techniques - Paper Example

The pairs show negative relationship because most of the values of X and Y are on opposite sides of the mean of X and Y respectively.

Question 6.7 (b)

Question 6.7 (c)

X Values

= 451

Mean = 50.111

S(X - Mx)^2= SSx = 1756.889 (See table below)

X X-Mx (X-Mx)^2

64 13.889 192.9043

40 -10.111 102.2323

30 -20.111 404.4523

71 20.889 436.3503

55 4.889 23.90232

31 -19.111 365.2303

61 10.889 118.5703

42 -8.111 65.78832

57 6.889 47.45832

Sum 451 1756.889

Mean 50.111

Y Values

= 687

Mean = 76.333

(Y - My)^2 = SSy = 858 (see table below)

Y Y-My (Y-My)^2

66 -10.333 106.7709

79 2.667 7.112889

98 21.667 469.4589

65 -11.333 128.4369

76 -0.333 0.110889

83 6.667 44.44889

68 -8.333 69.43889

80 3.667 13.44689

72 -4.333 18.77489

Sum 687 858

Mean 76.333

X and Y Combined

N = 9

(X - Mx) (Y - My) = Sxy = -1122.333

R Calculation

r = (X - Mx) (Y My) / SSx SSy

r = -1122.333 / 1756.889 858 = -0.9141

Question 6.10

Question 6.10 (a)

False.

Question 6.10 (b)

It may be true or false. Correlation does not imply causality (Martella, Nelson, Morgan, & Marchand-Martella, 2013). The negative relationship seen in children watching TVs may be due to other extraneous variables, such as laziness of the children who do not like studying but prefer watching television.

Question 6.10 (c)

True

Question 6.10 (d)

True

Question 6.10 (e)

False

Question 6.10 (f)

Can be true or false. Correlation does not imply causation (Warner, 2012). Therefore, no causal relationship can be established TV viewing, and academic performance can be inferred from correlation coefficient.

Question 6.11

It will not change the value of r.

r = (X Mx) (Y My) / SSx SSy = (X Mx) (Y My) / (X Mx)^2 S(Y My)^2

If the units of X are changed to X*, then those of Y to Y*, then

r = (X* Mx*) (Y My) / SSx* SSy* = (X* Mx*) (Y* My*) / (X* Mx*)^2 S(Y* My*)^2

Question 7.8

Question 7.8 (a)

Question 7.8 (b)

The X values are 5, 5, 2, 2, 3, 1, and 2. The sum, = (5 + 5 + 2 + 2 + 3 + 1+ 2) = 20

Mean = ( / n) = 20/6 = 2.85714

Therefore, S(X - Mx)^2= SSx = 14.8571

X X- Mx (X- Mx)^2 (SSx)

5 2.14286 4.591849

5 2.14286 4.591849

2 -0.85714 0.734689

2 -0.85714 0.734689

3 0.14286 0.020409

1 -1.85714 3.448969

2 -0.85714 0.734689

Sum 20 14.85714

Mean (Mx) 2.85714

The Y values are 4, 3, 2, 2, 2, 1, and 2.

The sum, = (4 + 3 + 2 + 2 + 2 + 1 + 2) = 16

Mean = ( / n) = 16/7 = 2.2857

Therefore, S(Y - My)^2= SSy = 5.428571 (see table below).

Y Y- My (Y- My)^2 (SSy)

4 1.7143 2.938824

3 0.7143 0.510224

2 -0.2857 0.081624

2 -0.2857 0.081624

2 -0.2857 0.081624

1 -1.2857 1.653024

2 -0.2857 0.081624

Sum 16 5.428571

Mean (Mx) 2.2857

X and Y Combined.

N = 7.

S(X - Mx)(Y - My) = Sxy = 8.28571

X- Mx Y- My (X-Mx)(Y-My)

2.14286 1.7143 3.673504898

2.14286 0.7143 1.530644898

-0.85714 -0.2857 0.244884898

-0.85714 -0.2857 0.244884898

0.14286 -0.2857 -0.040815102

-1.85714 -1.2857 2.387724898

-0.85714 -0.2857 0.244884898

8.285714286

Calculation of R

r = (X Mx) (Y My) / SSx SSy = (X Mx) (Y My) / (X Mx)^2 (Y My)^2 = 0.922613

b1 = Sxy / SSx = 8.286 / 14.857 = 0.5577

b0 = My - b1 Mx = (2.28571) (0.5577 2.85714) = 0.6923

y = b0 + b1 x = 0.6923 + 0.5577 x

Fit curve as shown in scatterplot above.

Question 7.8 (c)

Standard error of estimate = (SSy - b1 Sxy)/ (n - 2)) = (5.42857) (0.5577 8.28571) / (7 - 2)) = 0.40190

Question 7.8 (d)

y(2) = 0.6923 + 0.5577 2 = 1.8077

y(5) = 0.6923 + 0.5577 5 = 3.4808

Question 7.10

Question 7.10 (a)

True

Question 7.10 (b)

In this case, the dependence is co-variation. Therefore, it is false.

Question 7.10 (c)

False

Question 7.10 (d)

True

Question 7.13

If there was a random distribution of the offsprings. That is, the distribution is similar to the one of the present generation, then it will yield the same picture. Overall, the distribution will remain stable in the next generation.

References

Martella, R. C., Nelson, J. R., Morgan, R. L., & Marchand-Martella, N. E. (2013). Understanding and interpreting educational research. Guilford Press.

Warner, R. M. (2012). Applied statistics: from bivariate through multivariate techniques: from bivariate through multivariate techniques. Sage.