A confidence interval for a mean is a range of plausible values for an unknown population mean, based on data from a sample. It matters because a sample mean is only an estimate, and different random samples would give different results. The interval adds a margin of error around the sample mean to show how much uncertainty is reasonable.
A 95% confidence interval is commonly used because it balances precision with reliability.
Key Facts
- Point estimate for the population mean: x̄
- Standard error for a mean: SE = s / √n
- t-interval for a mean: x̄ ± t* (s / √n)
- Margin of error: ME = t* (s / √n)
- Degrees of freedom for a one-sample t-interval: df = n - 1
- A larger sample size usually makes the confidence interval narrower because SE decreases as n increases.
Vocabulary
- Confidence interval
- A range of values calculated from sample data that is used to estimate an unknown population parameter.
- Sample mean
- The average value x̄ computed from the observations in a sample.
- Margin of error
- The distance from the sample mean to either endpoint of the confidence interval.
- Critical t-value
- A multiplier from the t-distribution that sets how many standard errors are needed for a chosen confidence level.
- Standard error
- The estimated standard deviation of the sample mean, calculated as s / √n for a one-sample mean interval.
Common Mistakes to Avoid
- Saying there is a 95% chance that the population mean is inside this specific interval. The population mean is fixed, and the 95% refers to the long-run success rate of the method.
- Using z* when the population standard deviation is unknown. For most sample mean intervals with unknown σ, use a t* value with df = n - 1.
- Forgetting to divide the sample standard deviation by √n. The margin of error uses the standard error s / √n, not the raw sample standard deviation s.
- Interpreting a wider interval as a mistake. A wider interval can be correct when the data are more variable, the sample size is smaller, or the confidence level is higher.
Practice Questions
- 1 A sample of n = 25 students has mean study time x̄ = 6.8 hours and sample standard deviation s = 2.0 hours. Using t* = 2.064 for a 95% confidence interval, compute the interval for the population mean study time.
- 2 A sample of n = 40 batteries has mean lifetime x̄ = 18.6 hours and sample standard deviation s = 3.2 hours. Using t* = 2.023 for a 95% confidence interval, find the margin of error and the confidence interval.
- 3 Two studies estimate the same population mean. Study A uses n = 20 and Study B uses n = 200, with similar sample standard deviations and the same confidence level. Explain which interval should usually be narrower and why.