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Contingency Table and Probability

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Many people think that job satisfaction in the workplace is directly related to the annual income. To check the validity of this assumption a survey of 430 active working adults in the U.S was taken that relates their job satisfaction to their annual income. The results are shown in the contingency table below.

Annual Income Very Satisfied Moderately Satisfied Not Satisfied Total
$100,000 or more 32 4 4 40
$70,000 - $99,999 53 30 12 95
$40,000 - $69,999 84 80 26 190
Less than $40,000 25 37 43 105
Total 194 151 85 430

A) What proportion of the people survey are either 'very satisfied' or 'moderately satisfied'?
B) If one of the 430 people is selected at random, what is the probability that the person makes at least $70,000 annual income?
C) If one of the 430 people is selected at random, and given that the person is not 'very satisfied' with his/her job, what is the probability that the person makes $100,000 or more?

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Recovering the Cell Probabilities of a 2-by-2 Contingency Table

A 2-by-2 contingency table is Control on one axis and Experimental on the other. Control and Experimental are further dichotomized as Event and Non-event. The odds ratio (OR), number needed to harm (NNH), and absolute risk increase (ARI) are functions of the cell probabilities of a 2-by-2 contingency table, and conversely, the cell probabilities can be recovered given knowledge of these parameters. Recover the contingency table cell probabilities EE, EN, CE, and CN using OR, NNH, and ARI as knowns and the following definitions as necessary:

Experimental group (E), Control group (C), Events (E), Non-events (N), Total subjects (S), ES = EE + EN, CS = CE + CN, Event rate (ER), EER = EE / ES, CER = CE / CS, ARI = EER − CER, NNH = 1 / (EER − CER), and OR = (EE / EN) / (CE / CN). Also NNH = 1 + [CER x (OR-1)]/(1-CER) x (CER) x (OR-1).

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