Question

19 ERA 20 Win% 21 0.021 0.028 0.028 0.530 0.038 0.432 0.03 0.022 0.026 0.554 0.028 0.542 0.570 0.620 0.740 0.504 Compute the IQR for ERA. 23 24 25 26 27 28 29 30 31 32 Question 1A Question 1B Compute the covariance between ERA and win%. Question 1C Compute the correlation between ERA and win96. 34 36 Professor Smith took a sample of 211 students from her morning class and a sample of 160 students from her evening class. On Exam 1, all 37 those sampled from the morning class got a 64 while those sampled from the evening class averaged a 63 with a variance of 179. If 38 Professor Smith combines these two samples 39 40 42 43 45 46 47 48 49 Question 2 What is the combined Exam 1 average? Question 3A What is the combined Exam 1 median?

Answer 1

Solution

Let for convenience of representation, x denote ERA and y represent Win%

[NOTE: Formulas and Answers are given below. Detailed Working follow at the end.]

Q1 Part (a)

First quartile, Q1 = average of the second and the third observation in the ordered set

= (0.022 + 0.026)/2

= 0.024

Third quartile, Q3 = average of the sixth and the seventh observation in the ordered set

= (0.028 + 0.030)/2

= 0.029

IQR = Q3 – Q1

= 0.005 Answer

Q1 Part (b)

Covariance = (1/n)Σ(i = 1, n){(xi – Mean X)(yi – Mean Y)}

= - 0.00027   Answer

Q1 Part (c)

V(X) = (1/n)Σ(i = 1, n)(xi – Mean X)²

V(Y) = (1/n)Σ(i = 1, n)(yi – Mean Y)²

Correlation coefficient = Covariance/sqrt{V(X).V(Y)}

= - 0.00027/(0.0049 x 0.0843)

= - 0.6653    Answer

Details of Working

Let for convenience of representation, dxi = (xi – Mean X) and dyi = (yi – Mean Y)

i	x (ERA)	y(Win%)	dxi^2	dyi^2	(dxi)(dyi)
1	0.028	0.53	1.40625E-07	0.00099225	-1.2E-05
2	0.021	0.542	4.38906E-05	0.00038025	0.000129
3	0.038	0.432	0.000107641	0.01677025	-0.00134
4	0.03	0.57	5.64063E-06	7.225E-05	2.02E-05
5	0.028	0.62	1.40625E-07	0.00342225	2.19E-05
6	0.022	0.74	3.16406E-05	0.03186225	-0.001
7	0.026	0.554	2.64062E-06	5.625E-05	1.22E-05
8	0.028	0.504	1.40625E-07	0.00330625	-2.2E-05
Total	0.221	4.492	0.000191875	0.056862	-0.0022
Mean	0.0276	0.5615	2.39844E-05	0.00710775	-0.00027
Square Root of Variance			0.0049	0.0843

Ordered Set

0.021
0.022
0.026
0.028
0.028
0.028
0.03
0.038

DONE