8.2 A Single Population Mean using the Student t Distribution

Confidence Intervals and the Student's t-Distribution

Confidence intervals using the t-distribution

When you're estimating a population mean, you often don't know the true population standard deviation ($\sigma$). In practice, you almost never do. Instead, you use the sample standard deviation ($s$) as an estimate, and that extra uncertainty means you need the t-distribution instead of the normal (z) distribution.

The t-distribution is used when:

• The population standard deviation is unknown (you're using $s$ instead of $\sigma$)
• The underlying population is approximately normal, or the sample size is large enough for the Central Limit Theorem to apply

To build a confidence interval for a single population mean:

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

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

3. Look up the critical t-value ($t^*$) for your desired confidence level and degrees of freedom (using a t-table or technology)

4. Plug into the formula:

$\bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}}$

Where:

• $\bar{x}$ = sample mean
• $t^*$ = critical t-value
• $s$ = sample standard deviation
• $n$ = sample size

The piece $\frac{s}{\sqrt{n}}$ is the standard error of the mean. It estimates how much your sample mean typically varies from the true population mean. Multiply it by $t^*$ and you get the margin of error, which tells you how wide the interval stretches on each side of $\bar{x}$.

Example: Suppose you measure the heights of 20 students and get $\bar{x} = 66.5$ inches and $s = 3.2$ inches. For a 95% confidence interval with $df = 19$, the critical value is $t^* = 2.093$. The margin of error is $2.093 \cdot \frac{3.2}{\sqrt{20}} = 2.093 \cdot 0.716 \approx 1.50$ inches. Your 95% confidence interval is $(65.0, 68.0)$.

Degrees of freedom in the t-distribution

Degrees of freedom ($df$) represent the number of independent pieces of information in your sample that are free to vary. For a single sample mean, $df = n - 1$. You lose one degree of freedom because the sample standard deviation is calculated using $\bar{x}$, which "uses up" one piece of information.

Degrees of freedom control the shape of the t-distribution:

• Lower $df$ (small samples): thicker tails, which means more probability in the extremes. This produces larger critical values and therefore wider confidence intervals. The extra width accounts for the added uncertainty of estimating $\sigma$ with a small sample.
• Higher $df$ (larger samples): the tails thin out and the distribution looks more and more like the standard normal curve.

For example, a t-distribution with $df = 5$ has noticeably thicker tails than one with $df = 20$. By the time you reach $df = 100$ or so, the t-distribution is nearly identical to the standard normal.

T-distribution vs. normal distribution

Similarities:

• Both are symmetric and bell-shaped
• Both are centered at 0 (when standardized)
• Both are used for inference about population means

Differences:

• The t-distribution has thicker tails, meaning extreme values are more likely. This reflects the extra uncertainty from estimating $\sigma$ with $s$.
• The shape of the t-distribution changes depending on $df$, while the standard normal distribution has a single fixed shape.
• You use the t-distribution when $\sigma$ is unknown. You use the z (normal) distribution when $\sigma$ is known, which is rare in practice.

A common rule of thumb says to use the z-distribution when $n > 30$, but that's a simplification. The real deciding factor is whether you know $\sigma$. If you're using $s$, the t-distribution is technically correct regardless of sample size. With large $n$, though, the t and z values are so close that it barely matters.

Hypothesis testing with the t-distribution

While this section of the course focuses on confidence intervals, it's worth seeing how the t-distribution connects to hypothesis testing. The same logic applies: you're using $s$ instead of $\sigma$, so you use a t-statistic instead of a z-statistic.

Steps for a one-sample t-test:

1. State the null hypothesis ($H_0: \mu = \mu_0$) and alternative hypothesis
2. Choose a significance level ($\alpha$, commonly 0.05)
3. Calculate the test statistic:

$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

1. Find the p-value or compare the t-statistic to the critical value from the t-distribution with $df = n - 1$

2. If the p-value is less than $\alpha$ (or the test statistic falls in the rejection region), reject $H_0$

The t-statistic measures how many standard errors your sample mean is from the hypothesized value $\mu_0$. A larger absolute value means stronger evidence against the null hypothesis.