One-proportion z-test (Confidence interval)

Question: You are curious the how many freshmen students from Crammer Nation University joined Greek Life this year. You conduct a random sample of 50 randomly selected freshmen and find that 18 of them joined a Greek Life chapter. What is the 95% confidence interval?

As with all confidence interval problems, we're going to take the following steps to solve!

  • Step 1 - Calculate the confidence interval
  • Step 2 - Interpret the confidence interval

After doing the above steps, we're also going to cover a trick question you'll need to watch out for!

Step 1 - Calculate the confidence interval

We will use the following confidence interval formula to solve:

Let's go through each of the variables and plug them in!

Plugging in sample proportion (p-hat)

The prompt states that 18 out 50 freshmen students joined Greek Life...

You are curious the how many freshmen students from Crammer Nation University joined Greek Life this year. You conduct a random sample of 50 randomly selected freshmen and find that 18 of them joined a Greek Life chapter. What is the 95% confidence interval?

...therefore, since 18 / 50 = 0.36, we'll plug in 0.36 for p-hat!

Recognizing Margin of Error (MoE)

Our confidence interval will have upper and lower bounds from our sample proportion, 0.36.

That Margin of Error (MoE) will occur here...

...and will create the upper and lower bounds with the (±) here:

Determining the critical value (Z*)

When working with a Z* critical value (instead of a t*n-1 critical value), all we need to do is refer to the below table!

Confidence LevelZ* value
90%1.645
95%1.960
99%2.576

In our case, our confidence level is 95%...

You are curious the how many freshmen students from Crammer Nation University joined Greek Life this year. You conduct a random sample of 50 randomly selected freshmen and find that 18 of them joined a Greek Life chapter. What is the 95% confidence interval?

...therefore, we'll plug in 1.960 for Z*!

Plugging in sample proportion of failure (q-hat)

If you remember in What is a z-score, in relation to a sample proportion?, q represents the proportion of failure. In other words, it's the exact opposite of p.

q and p represent the population proportions of failure and success (respectively). q-hat and p-hat represent the sample proportions of failure and success (respectively).

Therefore, we can solve for the sample proportion of failure with the following equation:

q-hat = 1 - p-hat

Above, we found that p-hat equalled 0.36...

q-hat = 1 - 0.36

...therefore q-hat equals 0.64!

q-hat = 1 - 0.36 = 0.64

Let's go ahead and plug that in for q-hat!

Plugging in sample size (n)

The prompt states that the sample size was 50 freshmen students...

You are curious the how many freshmen students from Crammer Nation University joined Greek Life this year. You conduct a random sample of 50 randomly selected freshmen and find that 18 of them joined a Greek Life chapter. What is the 95% confidence interval?

...therefore we'll plug in 50 for n!

Solve for the confidence interval

When we solve this out, we get the following:

When we separate the "±" to create our lower bound (-) and upper bound (+)...

...it results in the following confidence interval!

Step 2 - Interpret the confidence interval

Before interpreting a confidence interval, I always recommend that you re-read the question to remind yourself what exactly you're interpreting. It's easy to get caught up in the math and forget.

You are curious the how many freshmen students from Crammer Nation University joined Greek Life this year. You conduct a random sample of 50 randomly selected freshmen and find that 18 of them joined a Greek Life chapter. What is the 95% confidence interval?

Now, to create our interpretation, let's refer back to the answer template referenced in What is a confidence interval?

We are confidence level % confidence that the true population mean vs. proportion description of what problem is finding is between lower CI bound and upper CI bound .

In our case, our answer would look like so:

Answer: We are 95% confidence that the true population proportion of Crammer Nation University freshmen who joined Greek Life this year is between 0.227 and 0.493.

Don't get tricked by this question!

Sometimes, you'll be a given a follow-up question to this interpretation like so:

Follow-up Question: Can you be sure that the true proportion of Crammer Nation University freshmen students who joined Greek Life is between 0.227 and 0.493?

This is a trick question! But... how so?

Since our 95% confidence interval resulted in the (0.227, 0.493), it's possible that the true population proportion lies between those values... however, it's not guaranteed. Remember, we're just 95% confident it's in that range... not 100% confident!

For this reason, we can not be sure that the true proportion of Crammer Nation University freshmen students who joined Greek Life is between 0.227 and 0.493.

Just because a value(s) falls inside the confidence interval doesn't mean the true population parameter is definitely that value(s)!

Don't get tricked by that on your exam!

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