8.1 A Single Population Mean Using the Normal Distribution

Confidence Intervals for a Single Population Mean

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Confidence intervals for population means

A confidence interval gives you a range of values that likely contains the true population mean, based on what you observed in a sample. The idea is straightforward: since a single sample mean probably won't land exactly on the population mean, you build a buffer around it using known information about sampling variability.

Before you can use the normal distribution to build this interval, two conditions must be met:

• The sample is a simple random sample from the population, ensuring the data is representative and unbiased.
• The population is normally distributed, or the sample size is large ($n \geq 30$). When $n \geq 30$, the Central Limit Theorem kicks in and guarantees the sampling distribution of $\bar{x}$ is approximately normal regardless of the population's shape.

The formula for a confidence interval is:

$\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$

where:

• $\bar{x}$ is the sample mean, your point estimate and the center of the interval
• $z^*$ is the critical value from the standard normal distribution, determined by your confidence level (e.g., $z^* = 1.645$ for 90%, $z^* = 1.96$ for 95%, $z^* = 2.576$ for 99%)
• $\sigma$ is the population standard deviation, which measures variability in the population (this formula assumes $\sigma$ is known)
• $n$ is the sample size

The quantity $\frac{\sigma}{\sqrt{n}}$ is called the standard error. It measures how much the sample mean tends to vary from sample to sample. Notice that as $n$ increases, the standard error shrinks, which makes the interval narrower. That's why larger samples give more precise estimates.

Steps to calculate a confidence interval:

1. Verify that both conditions (random sample and normality/large $n$) are met.
2. Identify $\bar{x}$, $\sigma$, and $n$ from the problem.
3. Determine the critical value $z^*$ based on the desired confidence level.
4. Calculate the margin of error: $z^* \frac{\sigma}{\sqrt{n}}$.
5. Compute the lower bound ($\bar{x} - \text{margin of error}$) and upper bound ($\bar{x} + \text{margin of error}$).

For example, suppose a random sample of $n = 50$ students has a mean test score of $\bar{x} = 78$, and the population standard deviation is $\sigma = 10$. For a 95% confidence interval:

• Standard error: $\frac{10}{\sqrt{50}} \approx 1.414$
• Margin of error: $1.96 \times 1.414 \approx 2.77$
• Interval: $(78 - 2.77,\ 78 + 2.77) = (75.23,\ 80.77)$

Interpretation of confidence intervals

Getting the interpretation right matters a lot on exams. A confidence interval provides a range of plausible values for the population mean based on your sample data.

The correct interpretation follows this template: "We are [confidence level]% confident that the true population mean of [context] is between [lower bound] and [upper bound]."

For the example above: "We are 95% confident that the true mean test score of all students is between 75.23 and 80.77."

What the confidence level actually means: 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. The confidence level describes the long-run success rate of the method, not any single interval.

Common misinterpretations to avoid:

• "There is a 95% probability that the population mean is in this interval." This is wrong. The population mean is a fixed number; it's either in the interval or it isn't. The 95% refers to the method's reliability across many samples.
• "95% of the data falls within this interval." The interval estimates the mean, not the spread of individual observations.

Sample size for margin of error

Sometimes you need to plan ahead: how large a sample do you need to achieve a specific margin of error? You can solve for $n$ algebraically from the confidence interval formula.

The margin of error ($E$) is the maximum expected difference between $\bar{x}$ and the true population mean. The formula for the required sample size is:

$n = \left(\frac{z^* \sigma}{E}\right)^2$

Steps to determine sample size:

1. Choose your desired confidence level and find the corresponding $z^*$.
2. Identify $\sigma$ (the population standard deviation). If unknown, use a reasonable estimate from prior studies or a pilot sample.
3. Decide on your desired margin of error $E$.
4. Plug into the formula and compute $n$.
5. Always round up to the next whole number. Rounding down would give you a margin of error slightly larger than what you wanted.

For example, if you want a 95% confidence interval ($z^* = 1.96$) with a margin of error of $E = 2$ and $\sigma = 10$:

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

You'd round up to $n = 97$.

Notice the trade-offs built into this formula: a higher confidence level (larger $z^*$) or a smaller margin of error (smaller $E$) both require a larger sample. More precision and more confidence cost more data.

Statistical Inference and Hypothesis Testing

Confidence intervals are one tool within the broader framework of statistical inference, which uses sample data to draw conclusions about populations.

The sampling distribution of $\bar{x}$ describes how the sample mean varies across all possible samples of size $n$. The confidence interval formula relies directly on the shape and spread of this distribution.

Hypothesis testing is a related technique. While a confidence interval estimates a parameter by giving a range of plausible values, a hypothesis test asks whether the data provide enough evidence to reject a specific claim about the parameter. The two approaches are connected: if a hypothesized value falls outside your confidence interval, you'd reject it at the corresponding significance level.

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