Confidence interval with one-mean t-test
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Scenario: You're going through fraternity rush at Crammer Nation University and hear that Delta Apple Pi brothers get a lot of Tinder matches. You take a random sample of 35 brothers and find a sample mean of 23.2 daily Tinder matches with a sample standard deviation of 3.2 matches. Based on this, find a 95% confidence interval for the true mean of daily Tinder matches of Sigma Apple Pi brothers.
| Clue | Insight |
|---|---|
| We're given a sample mean of 23.2. | We're working with means. |
| We're only given the sample standard deviation of 3.2, not the population standard deviation. | We'll have to settle for t-scores (instead of z-scores). |
| We need to find the 95% confidence interval for the true population mean. | We're working with confidence intervals. |
| Assumption | Validate |
|---|---|
| Sample is randomly selected from the population. | We see we took a "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 35 students is less than 10% of the entire population of the Delta Apple Pi chapter. ✅ |
| The sample size is greater than or equal to 30, or the population itself is normally distributed. | Our sample of 35 is greater than 30! ✅ |
x-bar is the sample mean that we've observed.
t*df is our t-score critical value.
s is the sample standard deviation.
n is the sample size.
x-bar = 23.2
s = 3.2
n = 35
t*df = ???
To find our critical value (t*n-1), we'll need...
df = n - 1
df = 35 - 1
df = 34
α = (1 - Confidence Level) / 2
α = (1 - 0.95) / 2
α = (0.05) / 2
α = 0.025
t*df = 2.032
CI = 23.2 ± 2.032(3.2 / √[35])
CI = 23.2 ± 2.032(3.2 / 5.916)
CI = 23.2 ± 2.032(0.541)
CI = 23.2 ± 1.099
CI = (22.101, 24.299)
Answer: We are 95% confident that the true population mean for daily Tinder matches for Delta Apple Pi brothers is between 22.101 and 24.299.