The t-distribution is a bell-shaped probability distribution used when estimating a population mean from a small sample. It matters because real data often come from samples where the population standard deviation is unknown. Compared with the standard normal distribution, the t-distribution has heavier tails, which allows for more uncertainty.
This makes confidence intervals wider and hypothesis tests more cautious when sample sizes are small.
The shape of a t-distribution depends on degrees of freedom, usually df = n - 1 for a one-sample mean problem. With low degrees of freedom, the curve is flatter in the center and thicker in the tails than the normal curve. As the sample size increases, the sample standard deviation becomes a better estimate of the population standard deviation, and the t-distribution approaches the standard normal distribution.
This is why t-methods are central tools for confidence intervals and tests about means.
Key Facts
- For a one-sample mean, degrees of freedom are df = n - 1.
- The t statistic for a sample mean is t = (xbar - μ0) / (s / sqrt(n)).
- Use the t-distribution when σ is unknown and the sample standard deviation s is used.
- A confidence interval for a mean is xbar ± t* s / sqrt(n).
- Smaller df gives heavier tails, so critical values t* are larger.
- As df increases, the t-distribution approaches the standard normal distribution.
Vocabulary
- t-distribution
- A family of bell-shaped probability distributions used to model standardized sample means when the population standard deviation is unknown.
- Degrees of freedom
- The number of independent pieces of information used to estimate variability, often df = n - 1 for a one-sample mean.
- t statistic
- A standardized value that measures how many estimated standard errors a sample mean is from a hypothesized population mean.
- Standard error
- The estimated standard deviation of a sampling distribution, equal to s / sqrt(n) for a sample mean.
- Critical value
- A cutoff value from a probability distribution that marks the boundary of a confidence interval or rejection region.
Common Mistakes to Avoid
- Using the normal distribution when σ is unknown and n is small. This is wrong because replacing σ with s adds extra uncertainty that the t-distribution accounts for.
- Forgetting to use df = n - 1 in a one-sample t problem. This is wrong because the sample mean uses one piece of information, leaving only n - 1 independent deviations.
- Thinking all t-distributions have the same shape. This is wrong because the curve changes with degrees of freedom, especially in the tails.
- Using s instead of s / sqrt(n) in the t statistic. This is wrong because the t statistic compares the mean to its standard error, not to the raw sample spread.
Practice Questions
- 1 A sample of n = 16 has mean xbar = 52 and standard deviation s = 8. Test against μ0 = 50 by computing the t statistic.
- 2 A sample of n = 25 has s = 10. Find the standard error of the mean and the degrees of freedom.
- 3 Explain why a t critical value for df = 5 is larger than a t critical value for df = 50 for the same confidence level.