Hypothesis test with one-proportion z-test
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Scenario: 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.
| Clue | Insight |
|---|---|
| 22 out of 50 students joined Greek Life, which is a proportion. | We're working with proportions here.test. |
| Proportions don't use t-tests. | We must use a z-score. |
| The scenario ends with "hypothesis test". | We're conducting a hypothesis test here. |
| Assumption | Validate |
|---|---|
| Sample is randomly selected from the population. | We can see "random sample" in the scenario. ✅ |
| The sample size (n) is less than or equal to 10% of the population size (N). | We can assume our sample of 50 students is less than 10% of the entire student population at Crammer Nation University. ✅ |
There are 10 successes and 10 failures in the sample OR np >= 10 and nq >= 10. | 22 students joined Greek Life ("successes"), leaving 50 - 22 = 28 who didn't ("failures"). Both are greater than 10! ✅ |
H0: p = 0.30
Ha: p ≠ 0.30
We're conducting a z-test, so our test statistic will be a z-score for proportions!
p-hat is the sample proportion we've observed.
p0 is the population proportion claimed by the null hypothesis (H0).
q0 is the population proportion of failure claimed by the null hypothesis (H0). It's essentially the opposite of p0.
n is the sample size.
p-hat = 22 / 50 = 0.44
p0 = 0.30
q0 = 1 - 0.30 = 0.70
n = 50
z = (0.44 - 0.30) / √[(0.30 x 0.70) / 50]
z = (0.14) / √[(0.21) / 50]
z = (0.14) / √[0.0042]
z = (0.14) / (0.065)
z = 2.15
p-value = 0.9842
But wait a second...
We're working with a two-tail test (so both the right and left tails of the distribution), since our alternative hypothesis uses ≠!
This means we need to double our p-value to reflect it upon the opposite tail of the distribution!
p-value = 0.0158 x 2
p-value = 0.0316
α = 0.05
Considering our p-value of 0.0316 is less than our alpha level of 0.05, this means we'll reject the null hypothesis!
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, which states the proportion of freshmen students at Crammer Nation University who joined Greek Life this year is not equal to 0.30.