Years of Experience Rookie 15 610 Over 10 Under 200 Over 300

Years of Experience Rookie 15 610 Over 10 Under 200 Over 300

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Years of Experience

Rookie

1-5

6-10

Over 10

Under 200

Over 300

200-300

1. Are the random variables of the joint probability distribution statistically independent? How can you tell?

2. Find the covariance of X and Y.

We have given the table of weight and year of experience.

And we have to check whether these two variables are independent or not.

This we can done by using chi-square distribution.

Here hypothesis for testing is,

H₀ : Weight and year of experience are independent.

H₁ : Weight and year of experience are not independent.

We doesn\'t give the value of level of significance () so we take as 0.05.

Applying the chi-square test for independence to sample data, we compute the degrees of freedom, the expected frequency counts, and the chi-square test statistic.

d.f. = (R-1)*(C-1)

where R = number of rows = 3

C = number of columns = 4

d.f. = (3-1)*(4-1) = 2*3 = 6

We calculate expected frequencies by using the formula,

E = (Row total)*(Column total) / sample size

We make the table of observad frequency and expected frequency as,

E11 = (18*8) /62 = 2.323

E12 = (30*8)/62 = 3.871

E13 = (12*8)/62 = 1.548

E14 = (2*8)/62 = 0.258

E21 = (18*41)/62 = 11.903

E22 = (30*41)/62 =19.839

E23 = (12*41)/62 = 7.935

E24 = (2*41)/62 = 1.323

E31 = (18*13)/62 = 3.774

E32 = (30*13)/62 = 6.290

E33 = (12*13)/62 = 2.516

E34 = (2*13)/62 = 0.419

Thus the Chi-square statistic value is 6.644

d.f. = 6 and level of significance = 0.05

critical value = 12.591

P-value = 0.354998

P-value > 0.05 also test statistic value is less than critical value.

So we fail to reject H₀ at 5% level of significance.

Conclusion : Weight and year of experience are independent.

The degree of association between the two variables can be assessed by a number of coefficients: the simplest is the phi coefficient defined by,

= sqrt(Chi-square statistic) / N

Where, N is the grand total of table = 62

= sqrt(6.644/62) = 0.3274

That means there is positive correlation between weight and years of experience.

That is there is some covariance value between these two variables.

O	E	(O-E)	(O-E)^2	(O-E)^2/E
3	2.323	0.677	0.458329	0.1973
5	3.871	1.129	1.274641	0.32928
0	1.548	-1.548	2.396304	1.548
0	0.258	-0.258	0.066564	0.258
11	11.903	-0.903	0.815409	0.068504
21	19.839	1.161	1.347921	0.067943
7	7.935	-0.935	0.874225	0.110173
2	1.323	0.677	0.458329	0.346432
4	3.774	0.226	0.051076	0.013534
4	6.29	-2.29	5.2441	0.83372
5	2.516	2.484	6.170256	2.452407
0	0.419	-0.419	0.175561	0.419
				6.644293

Years of Experience Rookie 1-5 6-10 Over 10 Under 200 Over 300 200-300 1. Are the random variables of the joint probability distribution statistically independe

Years of Experience

Weight (lb)

Rookie

$\"Y_{1}\"$

1-5

$\"Y_{2}\"$

6-10

$\"Y_{3}\"$

Over 10

$\"Y_{4}\"$

Under 200

$\"W_{1}\"$

Over 300

$\"W_{3}\"$

200-300

$\"W_{2}\"$