One-Sample t Inference: Distribution and Confidence Intervals

All flashcards

When is a one-sample Student's t-procedure used?

For inference about a population mean when $σ$ is unknown and $s$ estimates it.

What does a one-sample t-score measure?

The distance between $\bar{x}$ and $μ$ in estimated standard-error units: $t=(\bar{x}-μ)/(s/\sqrt{n})$.

What are the degrees of freedom for a one-sample t-procedure?

$df=n-1$, where $n$ is the sample size.

What is the error bound for a one-sample t-confidence interval?

$EBM=t_{α/2}s/\sqrt{n}$, the interval's margin of error.

How does a t-distribution change as its degrees of freedom increase?

It becomes more like the standard normal distribution, with less pronounced tails.

Quiz questions

1. Why does a one-sample t-procedure account for uncertainty differently from a z-procedure when $σ$ is unknown?Answer: It accounts for the added uncertainty from estimating $σ$ with $s$.
2. Which conditions should be considered when using a t-interval for a population mean, especially with a small sample?Answer: The sample is random, and the population is approximately normal.
3. For a two-sided 95% t-confidence interval, what area to the left of the positive critical value is used?Answer: 0.975
4. What does a 95% confidence level mean for a method of constructing intervals?Answer: Under the stated conditions, about 95% of intervals from repeated samples capture the true population mean.
5. What generally happens to a t-confidence interval when its confidence level is increased?Answer: It becomes wider because the critical t-value generally increases.