Confidence interval with two-mean t-test
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Scenario: Delta Apple Pi and Alpha Blueberry Pi are known across Crammer Nation University as the two fraternities that pull the most Tinder game. Considering that you're about to go through fraternity rush, you decide you want to determine the difference between each fraternity's average daily Tinder matches. You randomly sample brothers from each chapter and gather the following data:
| Delta Apple Pi | Alpha Blueberry Pi | |
|---|---|---|
| Sample size | 35 | 40 |
| Sample mean | 23.5 | 22.5 |
| Sample standard deviation | 5.7 | 3.9 |
Find the 95% confidence interval for the true difference in mean of daily Tinder matches between Delta Apple Pi and Alpha Blueberry Pi. The degrees of freedom is 60.
| Clue | Insight |
|---|---|
| We're given two sample means of 23.5 and 22.5. | We're working with two means. |
| We're given sample standard deviations of 5.7 and 3.9... not population standard deviations. | We need to settle for t-scores (instead of z-scores). |
| We need to find the 95% confidence interval for the true difference in population means between the chapters. | We're working with confidence intervals. |
| Assumption | Validate |
|---|---|
| Both samples are randomly selected from the population. | 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 sample sizes of 35 and 40 are less than 10% of the entire populations of Delta Apple Pi and Alpha Blueberry Pi (respectfully). ✅ |
| Both sample sizes (n1 and n2) are greater than or equal to 30, or the populations themselves are normally distributed. | Our sample sizes of 35 and 40 are both greater than 30! ✅ |
| Both samples are independent of each other. | We can assume that each chapter's average daily Tinder match data is independent of each other. (Ex: Ronnie getting a match doesn't impact Zach getting a match.) ✅ |
y-bar1 is the sample mean that we observed in the 1st sample.
y-bar2 is the sample mean that we observed in the 2nd sample.
t*df is our t-score critical value.
s1 is the sample standard deviation from the 1st sample.
s2 is the sample standard deviation from the 2nd sample.
n1 is the sample size from the 1st sample.
n2 is the sample size from the 2nd sample.
y-bar1 = 23.5
y-bar2 = 22.5
s1 = 5.7
s2 = 3.9
n1 = 35
n2 = 40
t*df = ???
To find our critical value (t*n-1), we'll need...
df = 60
α = (1 - Confidence Level) / 2
α = (1 - 0.95) / 2
α = (0.05) / 2
α = 0.025
t*df = 2.000
CI = (23.5 - 22.5) ± 2.000 √[(5.72 / 35) + (3.92 / 40)]
CI = (1.0) ± 2.000 √[(32.49 / 35) + (15.21 / 40)]
CI = (1.0) ± 2.000 √[(0.928) + (0.380)]
CI = (1.0) ± 2.000 √[1.308]
CI = (1.0) ± 2.000 (1.144)
CI = (1.0) ± 2.288
CI = (-1.288, 3.288)
Answer: We are 95% confident that the true difference in population mean between Delta Apple Pi's and Alpha Blueberry Pi's daily Tinder matches is between -1.288 and 3.288.
In other words... Delta Apple Pi may have a slight edge, because the majority of that interval is above 0. (This is NOT guaranteed though!)