Question

The fire department in a large city is examining its promotion policy to assess if there is evidence of age discrimination. A random sample of 248 promotion decisions over the past 5 years gives the following information:

39 or younger 40 or older
Promoted 38 46
Not Promoted 80 84

This data will be used to test if there is a relationship between age and promotion decision.

Required:
a. If there were no relationship between age and whether or not a firefighter was promoted, how many firefighters age 40 or older would we expect to be promoted?
b. What is the value of the Chi-square statistic for this test?

Answer 1

Answer:

A)

Hence the answer is 44.032.

B)

The value of $X^{2}$-test statistic for this test 0.280.

Step-by-step explanation:

A) The number of fire-fighter of age 40 or older to be promoted ith observed value 46 is to be expected 44.032 as calculated above,

So the answer is 44.032.

B) The value of $X^{2}$- test statistic is given by the formula,

$X^{2} =\sum \frac{\left ( O_{i}-E_{i} \right )^{2}}{E_{i}}\\X^{2} =0.97 +0.050+0.088+0.045\\X^{2} =0.280$

The value of $X^{2}$-test statistic for this test 0.280.

Answer 2

Using the Chisquare formula, the expected value and Chisquare statistic for the test would be 44.03 and 0.279 respectively

Recall :

The Chisquare formula is expressed thus :

• $X^{2} = \frac{(O_{I} - E_{I})^{2}}{E_{i}}$

1.)

(Row total × column total) / subtotal

(130×84)÷(84+164)

= 44.03

2.)

Actual Values:

38 ___ 46

80 ___ 84

Expected Values:

39.968 __ 44.032

78.032 __ 85.968

Using the Chisquared formula :

Chi-Squared Values:

0.0969 ___ 0.0879

0.0496 ___ 0.0450

Chi-Square = (0.0969 + 0.0496 + 0.0879 + 0.0450) = 0.279

Hence, the Chisquare statistic for the test is 0.279

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