Chapter 26, Problem 24E

a)

To determine

To find the probability that selected person is female.

Answer to Problem 24E

The probability that selected person is female = 0.1502

Explanation of Solution

Given:

  

Formula:

  $Probability=\frac{Numberofpossibleoutcomes}{Totalnumberofoutcomes.}$

The probability that selected person is female is,

  $P(Female)=\frac{5613}{37379}=0.1502$

b)

To determine

To find the probability that selected person is detective.

Answer to Problem 24E

The probability that selected person is detective = 0.1301

Explanation of Solution

Given:

  

Formula:

  $Probability=\frac{Numberofpossibleoutcomes}{Totalnumberofoutcomes.}$

The probability that selected person is detective is,

  $P(Detective)=\frac{4864}{37379}=0.1301$

c)

To determine

To find the number of female detectives would expect to have.

Answer to Problem 24E

The number of female detectives would expect to have = 730.4003

Explanation of Solution

Given:

  

Formula:

Expected value is

  $E=\frac{Rowtotal\times columntotal}{Grandtotal}$

The number of female detectives would expect to have:

  $E=\frac{5613\times 4864}{37379}=730.4003$

d)

To determine

To find which test is appropriate.

Answer to Problem 24E

Chi-square test for independence.

Explanation of Solution

Given:

  

There are two variables which want test verify either dependent or independent. Also, the counts of two variables are given. Hence, chi-square test of independence is appropriate to use.

e)

To determine

To state null and alternative hypotheses.

Answer to Problem 24E

  $H_0$ : Sex and Rank are independent.

  $H_a$ : Sex and Rank are not independent.

Explanation of Solution

Given:

  

The null and alternative hypotheses:

  $H_0$ : Sex and Rank are independent.

  $H_a$ : Sex and Rank are not independent.

f)

To determine

To check the assumptions and conditions.

Answer to Problem 24E

All the conditions satisfied.

Explanation of Solution

Given:

  

There are three main conditions of Chi-square test:

Counts, Independent and expected counts are at least 5.

The data has given counts so the condition of counts satisfied.

Assuming that officers at NYPD are independent, so condition of independent is satisfied.

The smallest expected count is

  $Female|Higherranks=\frac{228\times 5613}{37379}=34.2375$

Which is larger than 5 so all the conditions satisfied.

g)

To determine

To find the degrees of freedom.

Answer to Problem 24E

The degrees of freedom = 5

Explanation of Solution

Given:

  

Formula:

  $df=(r-1)(c-1)$

The number of rows = r = 6

The number of columns = c = 2

Therefore, the degrees of freedom are:

  $df=(6-1)\times (2-1)=5$

h)

To determine

To state the conclusion.

Answer to Problem 24E

There is sufficient evidence to reject the claim of independence.

Explanation of Solution

Given:

  ${\chi }^2$ = 290.1

P-value = 0.0001

Decision: The p-value< 0.05, reject H0.

Conclusion: There is sufficient evidence to reject the claim of independence.