7.2 Test your understanding
In this section you will try a problem on your own. Using the following activity, you can test your understanding of how to calculate the z-score for a problem related to the population and the sample proportion.
Activity 9 Testing proportion hypothesis
According to records, 70% of customers are willing to spend £15 on a 3D movie ticket. A ticket agent believes that this value is different. He surveys 200 individuals and found that 128 stated their willingness to spend a different amount on a ticket to a 3D movie. At the 95% confidence level, is there enough evidence to reject the null hypothesis?
Use the free response box below to show your calculations.
Discussion
Step 1: State the hypotheses.
The first step to tackle all the problems concerning hypothesis tests is to state the null and alternative hypotheses.
H0: 70% of customers are willing to spend £15 on a ticket to a 3D movie.
Ha: 70% of customers are not willing to spend £15 on a ticket to a 3D movie.
Or it can be written as
H0: p = 70% (or 0.7)
Ha: p ± 70%
Step 2: Use a z-score formula to calculate the z-score.
= sample (observed) proportion = = 0.64
p0 = population proportion = 0.70
n = sample size = 200
Using these values, you can calculate the z-score.
Step 3: Determine p-value using the Excel formula ‘NORM.S.DIST(z, cumulative)’
Judging from the null and alternative hypotheses, this is a two-tailed test.
Therefore, you need to divide the levels of significance by 2.
At 95% confidence level, the significance level equals 1 − 95%. Thus, α = 0.05
.
In order to reject the null hypothesis at 95% confidence levels in a two-tailed test, the p-value needs to be less than 0.025.
Given the z-score is negative, you can use Table 5 to determine the p-value. You can find the area left of z corresponding to a z-score of 1.85 is 0.0322.
Step 4: Make a decision based on the p-value.
The p-value from the calculation = 0.0322, which is ≥ 0.025.
Thus, you will not reject the null hypothesis that claims 70% of customers are willing to spend £15 on a ticket to a 3D movie.
Just a quick note: if this is not a two-tailed test (i.e. if it is a one tailed-test, associated with the left tail), you will not divide the significance levels by 2. In this situation, you will still use α = 0.05. So, as 0.0322 (p-value) < 0.05 (α), you would then reject H0.
This section aims to demonstrate how to use the z-score to make inferences about the population proportion based on sample data. When the population proportion is known, you can calculate the z-score to determine the likelihood of observing a particular sample proportion.