9.4 Hypothesis Test for Population Proportion Recall that the sample proportion is

9.4 Hypothesis Test for Population Proportion

Recall that the sample proportion is defined as:

The z-statistic for a population proportion is:

Hypothesis tests for population proportions use the z-table.

9.4.1 Left-Tailed Tests

Example 1 A manufacturer of alkaline batteries wants to know whether the defective rate of a shipment of it batteries is fewer than 5%. A sample of 300 batteries is randomly selected, and 10 defective batteries are found.

What is the population? What is the sample?

At α = 0.01 level of significance, determine whether there is evidence that the defective batteries in the shipment are less than 5%.

Solution

The population is all the batteries in the shipment. The sample is the 300 batteries selected.

(b)

-2.33

-2.33

•

•

α = α = 0.01

α = α = 0.01

-2.33

-2.33

-3 -2 -1 0 1 2 3

-3 -2 -1 0 1 2 3

z = -1.35

z = -1.35

The z-statistic is in the nonrejection region, so therefore there is no evidence that the defective batteries in the entire shipment are less than 5%.

9.4.3 Right-Tailed Tests

Example 2 Suppose a survey wants to determine the percentage (proportion) of people in a state who will vote a candidate. The survey selects a sample of 1000 voters and 550 of them say that they will vote the candidate.

What is the population? What is the sample?

At = 0.1 level of significance, determine whether there is enough evidence that the population proportion is more than 50% so that the candidate would win the election.

Solution

The population is all the voters in the state. The sample is the 1000 voters.

(b)

α = 0.1

α = 0.1

1.28

1.28

1.28

1.28

-3 -2 -1 0 1 2 3

-3 -2 -1 0 1 2 3

z = 3.16

z = 3.16

The z-statistic is in the rejection region, so there is evidence that the population proportion is more than 50%.

9.4.1 Two-Tailed Tests

Example 3 A hotel manager wants to know whether customer satisfaction is still the same as last year’s 87%. A survey of 1,100 customers shows that 945 of them are satisfied with the hotel.

What is the population? What is the sample?

At = 0.05 level of significance, determine whether there is evidence that the population proportion of customer satisfaction is different from last year’s 87%.

Solution

The population is all the customers of the hotel. The sample is the 1100 customers.

(b)

α/2 = 0.025

α/2 = 0.025

-1.96

-1.96

1.96

1.96

-1.96 1.96

-1.96 1.96

-3 -2 -1 0 1 2 3

-3 -2 -1 0 1 2 3

z = -0.99

z = -0.99

The z-statistic is in the nonrejection region, so there is no evidence that the population proportion of customer satisfaction is different from last year’s 87%.