7.1 Example: testing a proportion

‘A company believes that the percentage of people who subscribe to its mobile phone service is less than 25%. A sales manager disagrees with this because he conducted a study of 2000 people and found 540 of them subscribe to the company’s mobile phone services. At a 5% significance level, is there enough evidence to support this claim?’

Step 1: State the hypotheses.

The first step in addressing all the issues related to hypothesis tests is to formulate the null and alternative hypotheses. Unlike the example used in previous sections, which used the population mean (µ), here you need to use the population proportion (p).

H₀: the percentage of people that subscribe to the company’s mobile phone service is 25% or less.

H_a: the percentage of people that subscribe to the company’s mobile phone service is more than 25%.

This can also be written as:

H₀: p ≤ 25% (or 0.25)

H_a: p > 25%

Step 2: Use a z-score formula to calculate the z-score.

 = sample (observed) proportion = 0.27

 = population proportion = 0.25

 = sample size = 2000

Using these values, you can calculate the z-score:

Step 3: Determine the p-value using the z-scores in Table 5 or NORM.S.DIST() in Excel.

Judging from the null and alternative hypotheses, this is a one-tailed test on the right (upper) tail.

In Table 5, you can find that the area to the right of z corresponding to a z-score of 2.01 is 0.0222.

Thus, the p-value = 2.22%

Step 4: Make a decision based on the p-value.

According to the problem statement, you want to test the sales manager’s claim at 5% level of significance.

The p-value from the calculation = 2.22% < 5%

Thus, you reject the null hypothesis and accept the sales manager’s claim that the percentage of people that subscribe to the company’s mobile phone service is more than 25%.