Test for Homogeneity of Proportions Worksheet

**Page 3**

21.

A research surveys 125 people in each of four neighborhoods to determine the percentage of those whose annual income is greater than $50,000. The results are shown here. At $\alpha $ = 0.10, find the chi-square test value and test the claim that the proportions of those whose income is greater than $50,000 are equal in all four neighborhoods.

a. | 6.25, equal | ||

b. | 27.24, not equal | ||

c. | 6.25, not equal | ||

d. | 27.24, equal |

H

H

[Null and alternative hypotheses.]

The degrees of freedom is (2 - 1) × (4 - 1) = 3. The critical value at

[degrees of freedom = (row - 1) × (column - 1).]

The expected frequencies are computed as shown:

[Expected frequency =

The completed table is presented below with the expected frequencies, shown within brackets:

c

[O - Observed frequency, E - Expected frequency.]

c

[Expand and simplify.]

c

[Simplify.]

Since 27.24 > 6.251, the decision is to reject the null hypothesis.

Therefore, the proportions of those whose income is greater than $50,000 are not equal in all four neighborhoods.

Correct answer : (2)

22.

A research surveys 500 people in three different states to determine the proportion of the Republicans. The results are shown here.

At $\alpha $ = 0.10, find the chi-square test value and test the claim that the proportions of Republicans are equal in all three states.

State A | State B | State C | Total | |

Republicans | 280 | 244 | 226 | 750 |

Others | 220 | 256 | 274 | 750 |

Total | 500 | 500 | 500 | 1500 |

a. | 12.1, equal | ||

b. | 4.6, not equal | ||

c. | 12.1, not equal | ||

d. | 4.6, equal |

H

H

[Null and alternative hypotheses.]

The degrees of freedom is (2 - 1) × (3 - 1) = 2. The critical value at

[degrees of freedom = (row - 1) × (column - 1).]

The expected frequencies are computed as shown:

State A | State B | State C | Total | |

Republicans | 750 | |||

Others | 750 | |||

Total | 500 | 500 | 500 | 1500 |

[Expected frequency =

The completed table is presented below with the expected frequencies, shown within brackets:

State A | State B | State C | |

Republicans | 280(250) | 244(250) | 226(250) |

Others | 220(250) | 256(250) | 274(250) |

c

[O - Observed frequency, E - Expected frequency.]

c

[Expand and simplify.]

c

[Simplify.]

Since 12.10 > 4.605, the decision is to reject the null hypothesis.

Therefore, the proportions of Republicans are not equal in all three states.

Correct answer : (3)

23.

A survey was conducted in 3 towns to determine the proportions of people having pets in their home. The results are shown here.

At $\alpha $ = 0.05, find the chi-square test value and test the claim that the proportions are equal.

Town A | Town B | Town C | Total | |

Have pets | 32 | 45 | 40 | 117 |

Don't have pets | 53 | 40 | 45 | 138 |

Total | 85 | 85 | 85 | 255 |

a. | 4.07, equal | ||

b. | 4.07, not equal | ||

c. | 5.99, not equal | ||

d. | 5.99, equal |

H

H

[Null and alternative hypotheses.]

The degrees of freedom is (2 - 1) × (3 - 1) = 2. The critical value at

[degrees of freedom = (row - 1) x (column - 1).]

The expected frequencies are computed as shown:

Town A | Town B | Town C | Total | |

Have pets | 117 | |||

Don't have pets | 138 | |||

Total | 85 | 85 | 85 | 255 |

[Expected frequency =

The completed table is presented below with the expected frequencies, shown within brackets:

Town A | Town B | Town C | |

Have pets | 32(39) | 45(39) | 40(39) |

Don't have pets | 53(46) | 40(46) | 45(46) |

c

[O - Observed frequency, E - Expected frequency.]

c

[Expand and simplify.]

c

[Simplify.]

Since 4.07 < 5.99, the decision is not to reject the null hypothesis.

Therefore, the proportions of people having pets in their home are equal.

Correct answer : (1)

24.

Three batches of 100 animals were inoculated and were exposed to the infection of a disease. The results about the survival of the animals are tabulated as shown.

At $\alpha $ = 0.05, find the chi-square test value and test the claim that the proportions of animals which survived are equal for all the three batches.

Batch I | Batch II | Batch III | Total | |

Survived | 68 | 54 | 70 | 192 |

Dead | 32 | 46 | 30 | 108 |

Total | 100 | 100 | 100 | 300 |

a. | 6.59, equal | ||

b. | 9.21, equal | ||

c. | 6.59, not equal | ||

d. | 9.21, not equal |

H

H

[Null and alternative hypotheses.]

The degrees of freedom is (2 - 1) × (3 - 1) = 2. The critical value at

[degrees of freedom = (row - 1) × (column - 1).]

The expected frequencies are computed as shown:

Batch I | Batch II | Batch III | Total | |

Survived | 192 | |||

Dead | 108 | |||

Total | 100 | 100 | 100 | 300 |

[Expected frequency =

The completed table is presented below with the expected frequencies, shown within brackets:

Batch I | Batch II | Batch III | |

Survived | 68(64) | 54(64) | 70(64) |

Dead | 32(36) | 46(36) | 30(36) |

c

[O - Observed frequency, E - Expected frequency.]

c

[Expand and simplify.]

c

[Simplify.]

Since 6.59 < 9.210, the decision is not to reject the null hypothesis.

Therefore, the proportions of animals which survived in the three batches are equal.

Correct answer : (1)

25.

A survey was conducted to find out the proportions of boys who work in the age group 15 through 17. The results are shown.

At $\alpha $ = 0.05, find the chi-square test value and test the claim that the proportions of boys who work are equal.

15 year olds | 16 year olds | 17 year olds | Total | |

Work | 51 | 47 | 58 | 156 |

Don't work | 49 | 53 | 42 | 144 |

Total | 100 | 100 | 100 | 300 |

a. | 5.99, not equal | ||

b. | 2.48, not equal | ||

c. | 2.48, equal | ||

d. | 5.99, equal |

H

H

[Null and alternative hypotheses.]

The degrees of freedom is (2 - 1) × (3 - 1) = 2. The critical value at

[degrees of freedom = (row - 1) × (column - 1).]

The expected frequencies are computed as shown:

15 year olds | 16 year olds | 17 year olds | Total | |

Work | 156 | |||

Don't work | 144 | |||

Total | 100 | 100 | 100 | 300 |

[Expected frequency =

The completed table is presented below with the expected frequencies, shown within brackets:

15 year olds | 16 year olds | 17 year olds | |

Work | 51(52) | 47(52) | 58(52) |

Don't work | 49(48) | 53(48) | 42(48) |

c

[O - Observed frequency, E - Expected frequency.]

c

[Expand and simplify.]

c

[Simplify.]

Since 2.48 < 5.99, the decision is not to reject the null hypothesis.

Therefore, the proportions of boys who work are equal.

Correct answer : (3)