← Back to course

Confidence Intervals for a Mean

You’ll be able to

The one-sample t-interval

A confidence interval for a mean has the familiar form *x-bar ± t\ · (s/sqrt(n))*, where t\ is the critical value from the t-distribution with n − 1 degrees of freedom. The quantity s/sqrt(n) is the standard error of the mean, and t\*·(s/sqrt(n)) is the margin of error.

Conditions for t procedures

Three conditions: Random (data from an SRS or randomized experiment), 10% (n ≤ 0.10N when sampling without replacement), and Normal/Large Sample. The last is met if the population is Normal, or n ≥ 30 (CLT), or — for small samples — a graph of the data shows no strong skew or outliers. Always address the shape condition, especially when n is small.

One-sample t-interval for a mean
x-bar ± t\* · ( s / sqrt(n) ) with df = n − 1
t\* depends on the confidence level and df. For the same confidence level, t\* is larger than z\*, making t-intervals wider — the price of not knowing σ.
Worked example

A random sample of 25 batteries has x-bar = 40 hours and s = 5 hours. Construct a 95% confidence interval for the mean battery life (t\* = 2.064 for df = 24).

  1. 1.Conditions: random sample; assume batteries < 10% of all produced; n = 25, so check that a graph shows no strong skew/outliers (assume roughly symmetric).
  2. 2.Standard error: s/sqrt(n) = 5/sqrt(25) = 5/5 = 1.
  3. 3.Margin of error: t\*·SE = 2.064 × 1 = 2.064.
  4. 4.Interval: 40 ± 2.064 = (37.94, 42.06).
Answer: The 95% confidence interval is 40 ± 2.064 = (37.94, 42.06) hours. We are 95% confident the true mean battery life is between about 37.9 and 42.1 hours.
Checkpoint

For a one-sample t-interval based on n = 20 observations, how many degrees of freedom are used to find t\*?

On the exam

For small samples, you must address the shape condition explicitly: state that a dotplot/boxplot/stemplot of the data shows no strong skew or outliers, so it is plausible the population is approximately Normal. Skipping this on free response loses the conditions point.

Checkpoint

Holding everything else fixed, how does a 99% t-interval compare to a 95% t-interval for the same data?

Answer the 2 checkpoints as you read.

Sign in to save your progress