8.1 A Single Population Mean using the Normal Distribution

Confidence Intervals for a Single Population Mean

Confidence intervals for population means

A confidence interval gives you a range of values that likely contains the true population mean. Since you almost never know the actual population mean, you use sample data to estimate it.

The formula for a confidence interval is:

$\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$

Here's what each piece means:

• $\bar{x}$ is the sample mean, your best single estimate (called the point estimate)
• $z_{\alpha/2}$ is the critical value from the standard normal distribution, determined by your confidence level
• $\sigma$ is the known population standard deviation
• $n$ is the sample size

The critical value $z_{\alpha/2}$ changes depending on how confident you want to be. Common values:

• 90% confidence: $z_{\alpha/2} = 1.645$
• 95% confidence: $z_{\alpha/2} = 1.96$
• 99% confidence: $z_{\alpha/2} = 2.576$

The portion $z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$ is called the margin of error (E). It tells you how far above and below the sample mean your interval extends. Notice that as $n$ gets larger, $\sqrt{n}$ grows, which shrinks the margin of error. Bigger samples give you narrower, more precise intervals.

Example: Suppose you sample 49 students and find a mean test score of $\bar{x} = 82$, with a known population standard deviation of $\sigma = 7$. For a 95% confidence interval:

$E = 1.96 \cdot \frac{7}{\sqrt{49}} = 1.96 \cdot 1 = 1.96$

So the interval is $82 \pm 1.96$, or $(80.04,\ 83.96)$.

Interpretation of confidence intervals

Getting the interpretation right matters a lot on exams. A 95% confidence interval does not mean there's a 95% probability that the true mean falls in your specific interval. The true mean is a fixed number; it's either in the interval or it isn't.

The correct interpretation: if you repeated the sampling process many times and built a 95% confidence interval each time, about 95% of those intervals would contain the true population mean.

In context, you'd write something like: "We are 95% confident that the true average test score of the population lies between 80.04 and 83.96."

This approach relies on the sampling distribution of the sample mean being approximately normal. That's justified when:

• The population itself is normally distributed, or
• The sample size is large enough for the Central Limit Theorem to apply (generally $n \geq 30$)

Sample size for margin of error

Sometimes you need to figure out how many observations to collect before gathering data. If you want a specific margin of error $E$, rearrange the margin of error formula to solve for $n$:

$n = \left(\frac{z_{\alpha/2} \cdot \sigma}{E}\right)^2$

Steps to find the required sample size:

1. Choose your confidence level and find the corresponding $z_{\alpha/2}$
2. Identify the known population standard deviation $\sigma$
3. Decide on your desired margin of error $E$
4. Plug into the formula and solve
5. Always round up to the next whole number, since you can't sample a fraction of an observation

Example: You want a 95% confidence interval for a population mean with $\sigma = 10$ and a margin of error no larger than 2.

$n = \left(\frac{1.96 \cdot 10}{2}\right)^2 = (9.8)^2 = 96.04$

Round up: you need $n = 97$.

Notice the tradeoffs: a smaller margin of error or a higher confidence level both require a larger sample size.

Statistical inference and hypothesis testing

Confidence intervals and hypothesis tests are two sides of the same coin. Both use sample data to draw conclusions about population parameters.

A confidence interval estimates where the parameter is. A hypothesis test asks whether the parameter equals a specific value. For example, if a 95% confidence interval for a population mean does not contain the value $\mu_0$, you would reject $H_0: \mu = \mu_0$ at the $\alpha = 0.05$ significance level. The two methods will always agree when applied to the same data and significance level.

Note that this section uses the normal (z) distribution because the population standard deviation $\sigma$ is known. When $\sigma$ is unknown and you estimate it with the sample standard deviation $s$, you'll use the t-distribution instead, where degrees of freedom ($df = n - 1$) affect the shape of the distribution and the critical values.