Lecture 14
Inference for Population Means:
Confidence Intervals and t-Tests

ABD 3e Chapter 11

Chris Merkord

Learning Objectives

By the end of this lecture, you should be able to:

• Explain why the t-distribution is used instead of the normal distribution when the population standard deviation is unknown
• Describe how degrees of freedom influence the shape of the t-distribution
• Construct and interpret a confidence interval for a population mean
• State null and alternative hypotheses for a one-sample t-test
• Interpret the results of a t-test using t, degrees of freedom, p-value, and confidence interval

From Probability to Statistical Inference

• Previous lectures introduced probability distributions and hypothesis testing for categorical data and proportions.
• Today we apply the same inferential framework to means of numerical variables.
• The goal is to estimate and test hypotheses about a population mean using sample data.

Figure 1: Conceptual diagram showing how a population mean (parameter) and a sample mean (statistic) are obtained from a population and a sample. Source: Illustration generated by ChatGPT.

Sampling Distributions (Review)

• Statistics such as the sample mean vary from sample to sample due to sampling variability.
• The sampling distribution of the mean describes the distribution of possible sample means.
• If the population is normal, the sampling distribution of the mean is also normal.

Figure 2: Sampling distribution of the sample mean. Repeated random samples from the same population produce different sample means; the distribution of those means forms the sampling distribution. Source: GeeksforGeeks, “Sampling Distribution”.

Standard Error of the Mean (Review)

• The spread of the sampling distribution is measured by the standard error (SE) of the mean.
• Standard error quantifies how far sample means tend to vary from the population mean.
• Larger sample sizes reduce the standard error and improve precision of the estimate.

Figure 3: Sampling distributions of the mean penguin body mass for two sample sizes. Smaller samples (n = 10) produce more variable sample means than larger samples (n = 100).

When the Population SD Is Unknown

Population standard error

• The standard error of the mean (SE) describes the expected variability of sample means around the population mean.

• If the population standard deviation (SD) is known, the population SE is calculated using \(\sigma\).

• This formula describes the theoretical variability of the sample mean when the population variability is known.

Estimated standard error

• In practice, the population standard deviation \(\sigma\) is almost never known.

• Instead, we estimate variability using the sample standard deviation \(s\) calculated from the observed data.

• Replacing \(\sigma\) with \(s\) gives an estimated standard error of the mean.

\[ SE = \frac{\sigma}{\sqrt{n}} \]

\[ SE = \frac{s}{\sqrt{n}} \]

Standardizing the Mean (Review: Z statistic)

• In statistics, we often want to test:

  Does a sample provide evidence that a population mean differs from a hypothesized value.

• To evaluate how unusual a sample mean is, we measure its distance from the pop. mean in units of standard error.

• If the population standard deviation \(\sigma\) were known, we could use the Z statistic.

\[ Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \]

• The Z statistic measures how many standard errors the sample mean is from the population mean.

The Z distribution

Figure 4: Standard normal (Z) distribution used to measure how far a sample mean is from a hypothesized population mean in standard error units.

Z Scores vs Z Statistics

• The previous equation may look unfamiliar.

• In earlier lectures, we standardized individual observations using the population standard deviation \(\sigma\).

• When working with sample means, we instead standardize using the standard error of the mean.

• Sample means vary less than individual observations because they are averages of multiple values.

• The variability of sample means decreases as sample size increases, which is why the denominator includes \(\sqrt{n}\).

Situation	Formula	Term	Interpretation
Individual observation	\(Z = \frac{x - \mu}{\sigma}\)	Z score	Distance of an observation from the population mean in standard deviations
Sample mean	\(Z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}}\)	Z statistic	Distance of the sample mean from the population mean in standard errors

Replacing \(\sigma\) with \(s\) → the t statistic

• In practice, the population standard deviation \(\sigma\) is rarely known.

• Instead, we estimate population variability using the sample standard deviation \(s\).

• Substituting \(s\) for \(\sigma\) produces the t statistic.

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

• Because \(s\) varies from sample to sample, this introduces extra uncertainty in the standardized mean.

• As a result, the statistic follows a t distribution rather than the normal distribution.

The t distribution

Figure 5: Student’s t distribution for a small sample size, showing the distribution of the standardized mean when the population standard deviation is estimated from the sample.

The t distribution

• When the standard deviation is estimated from the sample, the standardized mean follows a t distribution.
• The t distribution is similar to the normal distribution but has heavier tails.
• Heavier tails reflect the additional uncertainty introduced when estimating \(\sigma\) with \(s\).
• The shape of the t distribution depends on the degrees of freedom.
• As sample size increases, the t distribution approaches the normal distribution.

Comparing the Z and t distributions

Figure 6: Comparison of the standard normal and Student’s t distributions. The t distribution has heavier tails because estimating the population standard deviation adds uncertainty. Figure generated using R.

Degrees of Freedom

1. The exact shape of the t distribution depends on the number of degrees of freedom.
2. For a sample mean the degrees of freedom are \(df = n - 1\).
3. Larger degrees of freedom produce a distribution closer to normal.

Figure 7: Student’s t distributions with different degrees of freedom. As degrees of freedom increase, the t distribution approaches the normal distribution. Figure generated using R.

What Are We Actually Estimating?

• The quantity of interest is the population mean \(\mu\).
• The sample mean \(\bar{x}\) provides an estimate of this parameter.
• Because of sampling variability our estimate always contains uncertainty.

Figure 8: A sample mean provides an estimate of the population mean. The confidence interval shows the uncertainty in that estimate.

Confidence Intervals for the Mean

• A confidence interval provides a range of plausible values for the population mean.
• The interval accounts for uncertainty caused by sampling variability.
• Is centered around the sample mean.
• Extends outward by a margin of error determined by the t distribution.
• Wider intervals indicate greater uncertainty in the estimate.

Figure 9: A confidence interval for a population mean constructed from a sample. The interval is centered on the sample mean and extends outward by a margin of error determined by the t distribution.

Calculating a Confidence Interval

• Confidence intervals for means are calculated using the t distribution.
• The interval combines the sample mean, the standard error, and a critical t value.
  • The critical value comes from the t distribution and depends on the chosen significance level ( \(\alpha\) ) and the degrees of freedom ( \(df = n - 1\) ) for the sample.
• The formula describes how far the interval extends from the sample mean.

\[ \bar{x} \pm t_{\alpha/2,df}\,SE \]

Figure 10: Annotated diagram showing components of the confidence interval formula. Source: Whitlock & Schluter 3e.

• The 2SE rule of thumb from lecture 6 derives from this equation
  • Uses a critical value of 2 (approximately 95% of sample means are within 2 SEs of the true mean).

Interpreting a Confidence Interval

• A 95% confidence interval captures the true mean in 95% of repeated samples.
• The interval represents uncertainty in estimating the population mean.
• Narrow intervals indicate more precise estimates.

Figure 11: Repeated 95% confidence intervals for a population mean. Each horizontal line is a confidence interval from one sample; the vertical red line shows the true population mean. Most intervals include the true value, but some do not.

Hypothesis Testing (Reminder)

• Earlier lectures introduced the general framework of hypothesis testing.
• We compare observed data with expectations under a null hypothesis.
• If the result is unlikely under the null hypothesis we reject it.

Figure 12: Probability distribution of the number of right-handed toads expected under the null hypothesis. The red tail regions represent outcomes at least as extreme as the observed value, which together form the p-value. Whitlock & Schluter 3e.

The one-sample t test

• The one-sample t test evaluates whether a sample mean differs from a hypothesized mean \(\mu_0\).
• Null hypothesis: \(H_0 : \mu = \mu_0\)
• Alternative hypothesis: \(H_A : \mu \ne \mu_0\)
• The difference between \(\bar{x}\) and \(\mu_0\) is scaled by the standard error.
• The resulting t statistic measures how extreme the observed difference is.

\[ t = \frac{\bar{x}-\mu_0}{SE} \]

Figure 13: Student’s t distribution with 4 degrees of freedom showing the rejection regions for a two-tailed test at α = 0.05. Each tail contains 2.5% of the probability, with critical values at 𝑡 = ± 2.78.

One-Sided and Two-Sided Tests

• A two-sided test evaluates whether the mean differs in either direction from \(\mu_0\).
• A one-sided test evaluates whether the mean is greater than or less than \(\mu_0\).
• Two-sided tests are typically used unless a directional hypothesis is justified.

(a) Two-tailed t test showing rejection regions in both tails (α = 0.05, 2.5% in each tail).

(b) One-tailed t test showing the rejection region in a single tail (α = 0.05).

Figure 14: Comparison of rejection regions for two-tailed and one-tailed t tests. In a two-tailed test, the significance level (α = 0.05) is split between both tails of the distribution, producing two rejection regions of 2.5% each. In a one-tailed test, the entire significance level is placed in a single tail, producing a single rejection region of 5%.

Assumptions of t Tests

• Data should represent a random sample from the population.
• The variable should be approximately normally distributed.
• Moderate deviations from normality are often tolerated with larger samples.

Figure 15: Distributions of stream chemistry variables from the Luquillo LTER dataset. Some variables are approximately symmetric, while others are strongly skewed, illustrating the diversity of real-world data distributions.

Sample Size and Interpretation

• Larger samples reduce the standard error and narrow confidence intervals.
• With large samples even small differences may become statistically significant.
• Statistical significance does not always imply biological importance.
  • Example: Average systolic blood pressure in a clinic sample (121 mmHg) differs significantly from the recommended value of 120 mmHg, but a 1 mmHg difference is unlikely to have meaningful consequences for patient health.
  • This is why you always report effect size (means, differences in means), not just p-values

Figure 16: Repeated 95% confidence intervals for a population mean at two sample sizes (n = 10 and n = 25). Each horizontal line is one CI; the vertical red line is the true population mean. Larger samples produce narrower intervals. Data Source: Palmer Penguins.

Reporting Results

• For a one-sample t-test:
  • Report the sample mean, standard deviation, and confidence interval.
  • Provide the t statistic, degrees of freedom, and p value.
• For an estimated mean:
  • Report the mean and confidence interval.
• Results should be interpreted in the biological context of the study.

Example:

I tested whether the average systolic blood pressure in my sample differed from the clinical reference value of 120 mmHg and found a statistically significant result (one-sample t-test, t₃₉=-2.45, two-sided p=0.019).

The mean blood pressure was 117.2 mmHg, a difference of −2.8 mmHg from the reference value, with a 95% confidence interval of −5.1 to −0.5 mmHg, indicating a small effect that is statistically detectable but unlikely to be clinically meaningful for individual patients.