8.2 A Single Population Mean Using the Student's t-Distribution

Confidence Intervals and the Student's t-Distribution

When the population standard deviation is unknown, you can't use a z-score to build a confidence interval. Instead, you use the Student's t-distribution, which accounts for the extra uncertainty that comes from estimating the standard deviation from your sample. The t-distribution is wider and has heavier tails than the normal distribution, giving you appropriately wider intervals when you have less information.

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Confidence Intervals Using the t-Distribution

The t-distribution is used to construct confidence intervals when the population standard deviation ($\sigma$) is unknown and must be estimated from the sample using $s$. While textbooks often cite $n < 30$ as the threshold for using $t$, in practice you should use the t-distribution any time $\sigma$ is unknown, regardless of sample size. For large $n$, the t- and z-intervals will be nearly identical anyway.

Steps to construct a confidence interval:

1. Calculate the sample mean ($\bar{x}$) and sample standard deviation ($s$) from your data.

2. Determine the degrees of freedom: $df = n - 1$.

3. Look up the critical t-value ($t^*$) for your desired confidence level (e.g., 95%) and your $df$, using a t-table or calculator.

4. Compute the margin of error: $E = t^* \cdot \frac{s}{\sqrt{n}}$

5. Build the interval: $\bar{x} \pm E$

The result is an interval $(\bar{x} - E,\; \bar{x} + E)$ that you are, say, 95% confident contains the true population mean $\mu$.

What affects the width of the interval?

• Sample size ($n$): Larger samples shrink the margin of error because $\sqrt{n}$ is in the denominator. More data means a more precise estimate.
• Confidence level: A higher confidence level (e.g., 99% vs. 90%) requires a larger $t^*$, which widens the interval. You're trading precision for greater confidence.
• Sample variability ($s$): More spread in your data increases the margin of error.

T-Scores vs. Z-Scores

Both t-scores and z-scores measure how far a sample mean is from a hypothesized population mean, in units of standard error. The key difference is which standard deviation you're using.

• Z-score: Used when $\sigma$ is known. The formula is $z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$.
• T-score: Used when $\sigma$ is unknown and you substitute the sample standard deviation $s$. The formula is:

$t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$

Where:

• $\bar{x}$ = sample mean
• $\mu$ = hypothesized population mean
• $s$ = sample standard deviation
• $n$ = sample size

Because $s$ is itself a random variable (it changes from sample to sample), the t-score has more variability than a z-score. That extra variability is exactly what the heavier tails of the t-distribution capture.

As $n$ grows large, $s$ becomes a very good estimate of $\sigma$, and the t-distribution converges to the standard normal (z) distribution.

Properties of the Student's t-Distribution

The t-distribution shares some features with the standard normal distribution but differs in important ways, especially at small sample sizes.

• Symmetric and bell-shaped, centered at zero, just like the standard normal.
• Heavier tails than the standard normal. This means extreme values are more likely under the t-distribution, which reflects the added uncertainty from estimating $\sigma$ with $s$.
• Defined by degrees of freedom ($df = n - 1$). The $df$ is the single parameter that controls the shape. Lower $df$ means heavier tails; higher $df$ means the distribution looks more and more like the standard normal.
• Standard deviation is greater than 1 for small $df$, and approaches 1 as $df$ increases.
• For $df > 30$ or so, the t-distribution is nearly indistinguishable from the standard normal. This is why older rules of thumb suggest switching to z at $n = 30$, but there's no harm in always using $t$ when $\sigma$ is unknown.

Type I and Type II Errors

These concepts connect confidence intervals to the broader framework of hypothesis testing.

• Type I error (false positive): Rejecting the null hypothesis when it is actually true. The probability of this is $\alpha$, your significance level. If you construct a 95% confidence interval, $\alpha = 0.05$, meaning there's a 5% chance the interval fails to contain the true mean.
• Type II error (false negative): Failing to reject the null hypothesis when it is actually false. The probability of this is $\beta$.

There's a tradeoff: making $\alpha$ smaller (to reduce false positives) increases $\beta$ (more false negatives), assuming sample size stays the same.

Statistical Power and Effect Size

Statistical power is the probability of correctly rejecting a false null hypothesis. It equals $1 - \beta$. Higher power means you're more likely to detect a real effect when one exists.

Three main factors influence power:

• Sample size: Larger $n$ reduces the standard error ($s / \sqrt{n}$), making it easier to detect differences. This is the factor researchers have the most control over.
• Effect size: The magnitude of the true difference between the hypothesized value and the actual population parameter. Larger effects are easier to detect. For example, detecting a 10-point difference in mean test scores is much easier than detecting a 1-point difference.
• Significance level ($\alpha$): A more stringent threshold (e.g., $\alpha = 0.01$ instead of $0.05$) requires stronger evidence to reject the null, which decreases power.

Effect size also helps you assess practical significance. A result can be statistically significant (small p-value) but have a tiny effect size, meaning the difference may not matter in the real world. Conversely, a meaningful effect might not reach statistical significance if the sample is too small. Both statistical and practical significance should be considered when interpreting results.