Hypothesis test with two-mean t-test
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Scenario: Delta Apple Pi claims that their brothers get more daily Tinder matches than Alpha Blueberry Pi brothers. You decide to test this claim by randomly sampling brothers from each chapter and gathering their average daily Tinder match data. You compile the following results:
| Delta Apple Pi | Alpha Blueberry Pi | |
|---|---|---|
| Sample size | 35 | 40 |
| Sample mean | 23.5 | 22.5 |
| Sample standard deviation | 5.7 | 3.9 |
Conduct a hypothesis test with α = 0.01 to assess this rumor, with degrees of freedom equaling 60.
| Clue | Insight |
|---|---|
| We have two sample means of 23.5 and 22.5. | We're working with two means here. |
| Our standard deviations of 5.7 and 3.9 are from the samples, not the populations. | We'll have to settle for a t-test (we can't take a z-test). |
| The scenario ends with "hypothesis test". | We're conducting a hypothesis test here. |
| Assumption | Validate |
|---|---|
| Both samples are randomly selected from the population. | We can see that we "randomly sampled brothers" in the scenario. ✅ |
| Both sample sizes (n1 and n2) is less than or equal to 10% of their respective population sizes. | We can assume our samples of 35 and 40 are less than 10% of the entire populations of the Delta Apple Pi and Alpha Blueberry Pi chapters (respectfully). ✅ |
| Both sample sizes (n1 and n2) are greater than or equal to 30, or the populations themselves are normally distributed. | Both our samples of 35 and 40 are greater than 30! ✅ |
| Both samples are independent of each other. | We can assume both chapter's average daily Tinder match data are independent of each other. ✅ (Ex: Timmy matching with Susie doesn't influence Hayden matching with Carly.) |
µ1 resembles Delta Apple Pi, µ2 resembles Alpha Blueberry Pi.
H0: µ1 - µ2 = 0
Ha: µ1 - µ2 > 0
We're conducting a t-test, so our test statistic will be a t-score!
y-bar1 is the sample mean we observed in the 1st sample.
y-bar2 is the sample mean we observed in the 2nd sample.
s1 is the sample standard deviation in the 1st sample.
s2 is the sample standard deviation in the 2nd sample.
n1 is the sample size in the 1st sample.
n2 is the sample size in the 2nd sample.
y-bar1 = 23.5
y-bar2 = 22.5
Δ0 = 0
s1 = 5.7
s2 = 3.9
n1 = 35
n2 = 40
t = [(23.5 - 22.5) - 0] / √[(5.72 / 35) + (3.92 / 40)]
t = [(1.0) - 0] / √[(32.49 / 35) + (15.21 / 40)]
t = [1.0] / √[(0.928) + (0.380)]
t = [1.0] / √[1.308]
t = [1.0] / 1.144
t = 0.874
df = 60
p-value > 0.10
But wait a second...
Our t-score sits somewhere greater than 0.10!
We're not done yet...
We're working with a one-tail test, specifically with the right-tail of the distribution, since our alternative hypothesis uses >!
p-value > 0.10
α = 0.01
Considering our p-value is something above 0.10, it's guaranteed to be greater than our alpha level of 0.01. This means we'll fail to reject the null hypothesis!
Answer: Since our p-value > 0.10 is greater than our alpha level of 0.01, we fail to reject the null hypothesis and do not have enough evidence to support the alternative hypothesis, which states the brothers of Delta Apple Pi have a mean daily Tinder match value greater than Alpha Blueberry Pi brothers.