Example: Gym membership Section
Research question: Are less than 50% of all individuals with a membership at one gym female?
A simple random sample of 60 individuals with a membership at one gym was collected. Each individual's biological sex was recorded. There were 24 females.
First we have to check the assumptions:
np = 60 (0.50) = 30
n(1-p) = 60(1-0.50) = 30
The assumptions are met to use the normal approximation method.
To perform a one sample proportion z test with summarized data in Minitab:
- In Minitab, select Stat > Basic Statistics > 1 Proportion
- Select Summarized data from the dropdown
- For number of events, add 24 and for number of trials add 60.
- Check the box next to Perform hypothesis test and enter 0.50 in the Hypothesized proportion box
- Select Options
- Use the default Alternative hypothesis setting of Proportion < hypothesized proportion value
- Use the default Confidence level of 95
- Select Normal approximation method
- Click OK and OK
The result should be the following output:
Event: Event proportion
Normal approximation is used for this analysis.
|N||Event||Sample p||95% Upper Bound for p|
|Null hypothesis||H 0: p = 0.5|
|Alternative hypothesis||H 1: p < 0.5|
We could summarize these results using the five-step hypothesis testing procedure:
\(np_0=60(0.50)=30\) and \(n(1-p_0)=60(1-0.50)=30\) both values are at least 10 so we can use the normal approximation method.
\(H_0\colon p = 0.50\)
\(H_a\colon p < 0.50\)
From output, \(z\) = -1.55
From output, \(p\) = 0.061
\(p \leq \alpha\), reject the null hypothesis
There is evidence that the proportion of women memberships at this gym is less than 50%.