t-Distribution
In short: t-distribution is a symmetric bell-shaped family with fatter tails than the normal curve, built for estimating or testing a mean once the true population spread has to be replaced by s.
AP StatisticsStatisticsAP Statistics Unit 4
What is t-Distribution?
The t-distribution is the normal curve’s cautious cousin. When you replace the unknown population standard deviation \(\sigma\) with the sample value \(s\), the standardized mean is a little more variable than a \(z\)-score, so the model needs fatter tails.
Its shape depends on the degrees of freedom, \(df = n – 1\). With small df the peak is lower and the tails are thicker, so a critical value \(t^*\) is larger than \(z^*\). As df grows, the curve approaches the standard normal. For a 95% interval, \(z^* = 1.960\), while \(t^* = 2.262\) at \(df = 9\) and \(2.064\) at \(df = 24\).
How to spot it
- An interval or test for a mean is requested, but the only spread you have is the sample’s own s.
- The sample is small, and the problem states that the population is roughly normal.
- A table has a column for degrees of freedom and rows of critical values.
- A graph shows a bell curve that is slightly flatter than the normal curve with longer tails.
The method
- Compute \(df = n – 1\) from the sample size, the first step for any t-distribution lookup.
- Look up the critical value \(t^*\) for your confidence level in the row for that df.
- Build the interval \(\bar{x} \pm t^*\frac{s}{\sqrt{n}}\).
- Check that the data are random and either \(n \ge 30\) or the sample shows no strong skew or outliers.
t-Distribution: solved example
Problem
Ten drone batteries run for an average of 23.4 hours with \(s = 3.1\). Assuming lifetimes are roughly normal, estimate the true average life with 95% confidence.
Solution
\(df = 10 – 1 = 9\), so \(t^* = 2.262\). The standard error is \(\frac{3.1}{\sqrt{10}} \approx 0.980\).
Margin of error: \(2.262 \times 0.980 \approx 2.217\).
Interval: \(23.4 \pm 2.217\), that is (21.18, 25.62) hours. Using \(z^* = 1.96\) would give an interval that is too narrow for such a small sample.
Try it
Question 1
For a 95% interval with 14 degrees of freedom, \(t^* = 2.145\). Is that larger or smaller than 1.96, and why?
Show the answer
Larger. With few degrees of freedom, the heavier tails require reaching farther out to capture the middle 95%.
Question 2
Why do statisticians use \(t\) instead of \(z\) for a mean?
Show the answer
Because \(\sigma\) is usually unknown, and estimating it by \(s\) adds extra variability that the t-distribution accounts for.
t-Distribution vs. Normal Distribution
The two curves look alike, but the normal curve has a single fixed shape. A t-distribution is a family indexed by degrees of freedom, with more probability in the tails. With df above 30 the difference is small, and the family approaches the normal curve in the limit.
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Whole dictionary: A–Z · AP Statistics · Statistics