One-proportion z-test (Hypothesis test)

Question: Crammer Nation University claims that 30% of their freshmen students joined a Greek Life chapter this year. You are curious if that's a truthful proportion, or if a different proportion of students joined a chapter. You collect a random sample of 50 freshmen students and find that 22 of them joined a chapter this year. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

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

  • Step 1 - State the hypothesis
  • Step 2 - Calculate the test statistic
  • Step 3 - Find the p-value
  • Step 4 - Make the concluding statement

Step 1 - State the hypotheses

Let's start with the null hypothesis (H0), then we'll move onto the alternative hypothesis (Ha).

Defining your null hypothesis (H0)

Here, we need to ask ourselves: what is the baseline claim that's being made about the population?

In the case of the situation above, it's that Crammer Nation University claims freshmen students joined Greek Life this year at a proportion of 0.30.

Crammer Nation University claims that 30% of their freshmen students joined a Greek Life chapter this year. You are curious if that's a truthful proportion, or if a different proportion of students joined a chapter. You collect a random sample of 50 freshmen students and find that 22 of them joined a chapter this year. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

We'd write that null hypothesis (H0) like so:

H0: p = 0.30

Essentially, what we're saying here is that the true population proportion for Greek Life involvement among Crammer Nation University freshmen...

H0: p = 0.30

...is equal to 0.30.

H0: p = 0.30

Defining your alternative hypothesis (Ha)

Here we need to ask ourselves: what's the claim that's being made that's different from the baseline claim about the population?

In the case of the situation above, it's that we're claiming that Crammer Nation University freshmen students joined Greek Life this year at a proportion not equal to 0.30.

Crammer Nation University claims that 30% of their freshmen students joined a Greek Life chapter this year. You are curious if that's a truthful proportion, or if a different proportion of students joined a chapter. You collect a random sample of 50 freshmen students and find that 22 of them joined a chapter this year. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

We'd write that alternative hypothesis (Ha) like so:

H0: p = 0.30
Ha: p ≠ 0.30

Essentially what we're saying here is that the true population proportion for Greek Life involvement among Crammer Nation University freshmen...

H0: p = 0.30
Ha: p ≠ 0.30

...is actually not equal to 0.30.

H0: p = 0.30
Ha: p ≠ 0.30

Step 2 - Calculating your test statistic

Before even beginning to calculate our test statistic, we have to check out assumptions!

Check your assumptions!

We're working with a sample proportion here, so in accordance with Assumptions for sampling distributions, that means we must check the following assumptions:

1. Sample is randomly selected from the population
2. The sample size (n) is less than or equal to 10% of of the population size N
3. There are 10 successes and 10 failures in the sample OR np >= 10 and nq >= 10

Concerning #1, we're collecting a random sample from the population:

Crammer Nation University claims that 30% of their freshmen students joined a Greek Life chapter this year. You are curious if that's a truthful proportion, or if a different proportion of students joined a chapter. You collect a random sample of 50 freshmen students and find that 22 of them joined a chapter this year. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

Concerning #2, we can assume that 50 freshmen students is less than 10% of the entire freshmen student body at Crammer Nation University.

Crammer Nation University claims that 30% of their freshmen students joined a Greek Life chapter this year. You are curious if that's a truthful proportion, or if a different proportion of students joined a chapter. You collect a random sample of 50 freshmen students and find that 22 of them joined a chapter this year. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

Concerning #3, we have 22 "successes" (freshmen students who joined Greek Life) and therefore have 50 - 22 = 28 "failures" (freshmen students who didn't join Greek Life).

Therefore, all of our assumptions are passed! Now we can move onto calculating our test statistic!

Recognizing we'll use z-score

Remember, you'll only use t-score if you're dealing with sample means! Therefore, we know we will be using z-score here, which will be computed with the following formula:

This looks similar to the regular z-score formula... what's with "p0" and "q0"

You'll notice this formula is very similar to the formula for z-scores in What is a z-score, in relation to a sample proportion?...

...but now it's using p0 and q0 instead of p and q.

Long story short, that's because with hypothesis tests, we use the "0" (that little "0" is often called "knot") to signify that we don't know for sure that it's the true population proportion for p and q. It's the claimed population proportion that's being tested within our hypothesis test!

Plugging in the sample proportion (p-hat)

Based on the prompt, the sample proportion was 0.44 (because 22 / 50 = 0.44)...

Crammer Nation University claims that 30% of their freshmen students joined a Greek Life chapter this year. You are curious if that's a truthful proportion, or if a different proportion of students joined a chapter. You collect a random sample of 50 freshmen students and find that 22 of them joined a chapter this year. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

...therefore, we'll plug in 0.44 for p-hat.

Plugging in the claimed population proportion (p0)

Based on the prompt, the sample proportion was 0.44 (because 22 / 50 = 0.44)...

Crammer Nation University claims that 30% of their freshmen students joined a Greek Life chapter this year. You are curious if that's a truthful proportion, or if a different proportion of students joined a chapter. You collect a random sample of 50 freshmen students and find that 22 of them joined a chapter this year. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

...therefore, we'll plug in 0.44 for p-hat.

Plugging in the claimed population proportion of failure (q0)

Next, for our population proportion of failure (q0), we'll do what we did in What is a z-score, in relation to sample proportions? and utilize the following formula:

q0 = 1 - p0

Since p0 equals 0.30...

q0 = 1 - 0.30

...this results in q0 equalling 0.70...

q0 = 1 - 0.30 = 0.70

...so we'll plug in 0.70 for q0!

Plugging in the sample size (n)

Lastly, the prompt states that the sample size is 50...

Crammer Nation University claims that 30% of their freshmen students joined a Greek Life chapter this year. You are curious if that's a truthful proportion, or if a different proportion of students joined a chapter. You collect a random sample of 50 freshmen students and find that 22 of them joined a chapter this year. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

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

Solving for z-score

When we solve this out, it results in a z-score of 2.15!

Step 3 - Find your p-value

First, let's determine if we're doing a one-tail or two-tail test, and then let's find our p-value in the z-table.

One-tail or two-tail?

In What is a hypothesis test?, we declared the following:

If your alternative hypothesis has ">" or "<", that means it's a one-tail test.
If your alternative hypothesis has "", that means it's a two-tail test.

Since our alternative hypothesis uses a "≠", that means we're dealing with a two-tail test.

With two tail tests, that means our alpha level of 0.05...

Crammer Nation University claims that 30% of their freshmen students joined a Greek Life chapter this year. You are curious if that's a truthful proportion, or if a different proportion of students joined a chapter. You collect a random sample of 50 freshmen students and find that 22 of them joined a chapter this year. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

...will be split among the two tails.

0.05 / 2 = 0.025

Finding our p-value

Knowing that our z-score is 2.15, all we need to do is go to our z-table...

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...find "2.1" in the left-hand column (representing 2.15)...

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...and then "0.05" in the top row (representing 2.15)...

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...to locate our p-value of 0.9842!

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Visualizing our p-value

We need to remember that the z-table displays the area to the left of your z-score...

...therefore this p-value of 0.9842 can be understood visually like so:

To find the p-value to the right of our p-hat value, we must subtract 0.9842 from 1.

1.00 - 0.9842 = 0.0158

Duplicating our p-value on both tails

Remember, we're dealing with a two-tail test.

That means that we must reflect our p-hat value of 0.44...

...onto the left-tail of our sampling distribution.

This results in a final p-value of 0.0316! We can also see that visually, this fits in our alpha level range on both tails of our sampling distribution!

Step 4 - Make your concluding statement

Your concluding statement is going to center around the alpha level declared in the problem. In most cases, that alpha level will be 0.05. Each problem should explicitly state the alpha level. In our problem, it's 0.05.

Crammer Nation University claims that 30% of their freshmen students joined a Greek Life chapter this year. You are curious if that's a truthful proportion, or if a different proportion of students joined a chapter. You collect a random sample of 50 freshmen students and find that 22 of them joined a chapter this year. Provide support for your claim using a hypothesis test with an alpha level of 0.05.

As stated in What is a hypothesis test?...

- If the p-value is below the alpha level, then we reject the null hypothesis
- If the p-value is above the alpha level, then we fail to reject the null hypothesis.

Our p-value of 0.0316 is below our alpha level of 0.05, therefore we'll reject our null hypothesis! The chances of our sample occurring are so low, that they are statistically significant enough to support the alternative hypothesis.

Applying the z-score answer statement template

If you remember in What is a hypothesis test?, we gave the following answer template:

Since our p-value of p-value is less / greater than our alpha level of alpha level value, we reject / fail to reject the null hypothesis and do / don't have enough evidence to support the alternative hypothesis, implying that description of alternative hypothesis.

Applied to our question, this would give us the following answer to our original question!

Answer: Since our p-value of 0.0316 is less than our alpha level of 0.05, we reject the null hypothesis and do have enough evidence to support the alternative hypothesis, implying that the proportion of freshmen students at Crammer Nation University who joined Greek Life this year is not equal to 0.30.

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