Before we can construct a confidence interval for a proportion we must first determine if we should use the exact method or the normal approximation method. Recall that if \(np \geq 10\) and \(n(1-p) \geq 10\) then the sampling distribution can be approximated by a normal distribution. Since we don't have the population proportion (\(p\)), we using \(\widehat p\) as an estimate. Note that \(n\widehat p\) is the number of successes in the sample and \(n(1-\widehat p)\) is the number of failures in the sample.
If this assumption has not been met, then the sampling distribution is constructed using a binomial distribution which Minitab refers to as the "exact method."
To check this assumption we can construct a frequency table. You first learned how to construct a frequency table in Lesson 188.8.131.52.1 of these online notes. Here is another example:
Minitab® – Frequency Tables
To create a frequency table of dog ownership in Minitab:
- Open the data set:
- From the toolbar in Minitab, select Stat > Tables > Tally Individual Variables
- Double click the variable Dog in the box on the left to insert the variable into the Variable box
- Under Display, choose Counts
- Click OK
This should result in the following frequency table:
From the frequency table above we can see that there were at least 10 "successes" and at least 10 "failures" in the sample. In this example a success is defined as answering "yes" to the question "do you own a dog?" A failure is defined as answering "no." Because both \(n \widehat p \geq 10\) and \(n(1- \widehat p) \geq 10\), the normal approximation method may be used. In Minitab, the exact method is the default method. If there are at least 10 successes and at least 10 failures, then you need to change the method to the normal approximation method.
Minitab® – Confidence Interval for a Proportion (Normal Approximation)
To create a 95% confidence interval of dog ownership using the normal approximation method in Minitab:
- Open the data set: fall2016stdata.mpx
- In Minitab, select Stat > Basic Statistics > 1-Proportion
- In this case we have our data in the Minitab worksheet so we will use the default One or more samples each in a column.
- Double click the variable Dog in the box on the left to insert the variable into the box.
- Select Options
- The default Confidence level is 95
- Change the Method to Normal approximation because the assumption of \(n \widehat p \geq 10\) and \(n(1- \widehat p) \geq 10\) has been met
- Click OK
This should result in the following output:
Event: Dog = Yes
p: proportion where Dog = Yes
Normal approximation is used for this analysis.
|N||Event||Sample p||95% CI for p|
What if the assumption is not met? Section
If the number of successes or the number of failures in the sample is less than 10, then the exact method should be used instead of the normal approximation method. In Minitab, this means that in step 8 above the default setting of Exact method should not be changed.
If you do not have a Minitab worksheet filled with data concerning individuals, but instead have summarized data (e.g., the number of successes and the number of failures), you would not load the data set, but in step 3 you would select Summarized data. For Number of events, enter the number of successes (i.e., \(n \widehat p\)) and for Number of trials enter the total sample size (i.e., \(n\)).