Inference About a Mean: Basic Biostatistics Study Guide

Chapter 11 Overview: Inference About a Mean

• Main Subject: Basic Biostatistics - Statistics for Public Health Practice.

• Author: B. Burt Gerstman.

• Date: 5/8/2026.

• Contents of Chapter 11:     * 11.1 Estimated Standard Error of the Mean.     * 11.2 Student’s t Distribution.     * 11.3 One-Sample t Test.     * 11.4 Confidence Interval for $\mu$.     * 11.5 Paired Samples.     * 11.6 Conditions for Inference.

Estimated Standard Error and the Transition to Student's t Procedures

• When $\sigma$ is Unknown:     * In previous chapters, it was assumed that the population standard deviation ($\sigma$) was known before data collection, allowing for the use of $z$ procedures to infer the population mean ($\mu$).     * When $\sigma$ is NOT known, the researcher must calculate the sample standard deviation ($s$) and use it to calculate the standard error.

• The Impact of Additional Uncertainty:     * Using the sample standard deviation ($s$) instead of the population parameter ($\sigma$) introduces additional uncertainty into statistical inferences.     * Because of this uncertainty, $z$ procedures cannot be used. Instead, researchers rely on Student’s t procedures.     * The Standard Normal distribution does not fit the data well when $\sigma$ is estimated from the sample.

• Historical Context: Student's t procedures were developed by William Sealy Gosset ($1876$–$1937$).

T-score vs. Z-score Usage Criteria

• A t-score should be used instead of a z-score when:     * The sample size ($n$) is below $30$.     * The population standard deviation ($\sigma$) is unknown and must be estimated from the sample data using $s$.

Characteristics of Student’s t Distributions

• Probability Distributions and Degrees of Freedom: t distributions are a family of probability distributions identified by a parameter called degrees of freedom ($df$).

• Shape Comparison to Z:     * The t distribution is similar in shape to the Standard Normal "Z" distribution (bell-shaped and symmetrical around $0$).     * The t distribution has broader (heavier) tails compared to the Z distribution.

• Effect of Degrees of Freedom:     * As the $df$ increases, the tails of the t distribution become "skinnier" (thinner).     * As $df$ increases, the t distribution becomes more like the Z distribution.     * A t distribution with infinite degrees of freedom ($df = \infty$) is identical to the Standard Normal Z distribution.

The t-Distribution Table (Table C)

• Structure:     * Rows correspond to the degrees of freedom ($df$).     * Columns correspond to selected right-tail (greater-than) probabilities ($p$).     * Entries in the table are values ($t$) on a t-distribution.

• Notation: $t_{cum_prob,df} = t \text{ value}$. For example, $t_{.975, 9} = 2.262$.

• Sample Data from Table:     * For $df = 1, p = 0.05$, $t = 6.313752$.     * For $df = 10, p = 0.025$, $t = 2.22814$.     * For $df = 30, p = 0.0005$, $t = 3.6460$.     * For infinite $df$ ($\infty$) at $95\%$ Confidence Interval level (which corresponds to $p = 0.025$ for a two-tailed tail), the value is $1.95996$.

The t-Test Fundamentals

• Statistical Significance: The t statistic is a measure of how likely it is that an observed difference in means is statistically significant.

• Comparison: Like all test statistics, the calculated t value is compared to a critical value ($t_{critical}$).     * The value of $t$ is calculated from the sample data.     * The value of $t_{critical}$ is determined by the selected Alpha ($\alpha$ or significance level) and the appropriate t-distribution based on $df$.

• Interpreting the t-Value: A larger value for $t$ makes it more likely to exceed $t_{critical}$, which increases the likelihood that there is a statistically significant difference in the means.

• Basic Formula:     $t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$

Classifications of t-Tests

• 1-Sample t-test:     * Mean 1: Sample from a population or process.     * Mean 2: Specified mean (a target, an estimate, or a historical value).

• 2-Sample t-test:     * Mean 1: Sample from a population or process.     * Mean 2: Sample from a different population or process.

• Paired t-test:     * Mean 1: One half of a sample of paired data (e.g., score before training).     * Mean 2: The other half of the sample of paired data for the same test subjects (e.g., score after training).

Degrees of Freedom ($df$)

• Single Sample t-test: $df = n - 1$, where $n$ is the sample size.

• Two-Sample t-test: $df = n_1 + n_2 - 2$.

One-Sample t-Test: Objective and Conditions

• Objective: To test a claim about a population mean ($\mu$).

• Conditions:     * Simple Random Sample (SRS).     * Normal population or a "large sample."

Hypothesis Statements for One-Sample t-Test

• Null Hypothesis ($H_0$): $H_0: \mu = \mu_0$, where $\mu_0$ represents the population mean expected under the null hypothesis.

• Alternative Hypotheses ($H_a$):     * $H_a: \mu < \mu_0$ (One-sided, left-tailed).     * $H_a: \mu > \mu_0$ (One-sided, right-tailed).     * $H_a: \mu \neq \mu_0$ (Two-sided).

Case Study: SIDS Birth Weights Example

• Research Question: Do Sudden Infant Death Syndrome (SIDS) babies have lower average birth weights than the general population mean ($\mu$) of $3300\,g$?

• Hypotheses:     * $H_0: \mu = 3300$.     * $H_a: \mu < 3300$ (one-sided) or $H_a: \mu \neq 3300$ (two-sided).

• Data ($n = 10$ SRS of birth weights in grams): $2998, 3740, 2031, 2804, 2454, 2780, 2203, 3803, 3948, 2144$.

• Calculated Statistics:     * Sample Mean ($\bar{x}$): $2890.5$.     * Standard Deviation ($s$): $720$.     * Standard Error ($SE_{\bar{x}}$): $\frac{s}{\sqrt{n}} = \frac{720}{\sqrt{10}} = 227.68$.

• Test Statistic Calculation:     * $t_{stat} = \frac{\bar{x} - \mu_0}{SE_{\bar{x}}} = \frac{2890.5 - 3300}{227.7} = -1.80$.     * $df = n - 1 = 10 - 1 = 9$.

• P-value Determination via Table C:     * For $|t_{stat}| = 1.80$ with $9\,df$:     * Looking at the row for $df=9$, $1.80$ falls between the values for probabilities $0.05$ and $0.10$.     * Upper-tail probabilities: $t=1.383 \rightarrow p=0.10$; $t=1.833 \rightarrow p=0.05$.     * One-tailed P-value: $0.05 < P < 0.10$.     * Two-tailed P-value: $0.10 < P < 0.20$.

• Interpretation:     * Testing $H_0: \mu = 3300\,g$.     * The two-tailed $P > .10$.     * Conclusion: There is weak evidence against the null hypothesis. The sample mean ($2890.5$) is not significantly different from $3300$.

Confidence Intervals for $\mu$

• Definition: A measure of the reliability of a result. A confidence level of $95\%$ (or $0.95$) means there is a probability of at least $95\%$ that the result is reliable.

• Calculation for SIDS Data (95% CI):     * $n = 10$, $df = 9$.     * $t_{n-1, 1-\frac{\alpha}{2}} = t_{9, .975} = 2.262$.     * $95\% \text{ CI for } \mu = \bar{x} \pm (t_{9, .975} \times SE_{\bar{x}})$.     * Calculation: $2890.5 \pm (2.262 \times 227.68) = 2890.5 \pm 515.1$.     * Interval: $(2375 \text{ to } 3406)\,g$.

• Interpretation: We are $95\%$ confident that the population mean $\mu$ is between $2375$ and $3406\,g$.

The Normality Condition for Inference

• Guidelines for using t procedures:     * Use when the population is Normal.     * Use when the population is symmetrical and $n \geq 10$.     * Use when the population is skewed and $n \geq \text{approximately } 45$ (the specific requirement depends on the severity of the skew).

Sample Size and Power

• Methods Include:     1. Calculating $n$ required to achieve a specific margin of error ($m$) when estimating $\mu$.     2. Calculating $n$ required to test $H_0$ with a desired power ($1-\beta$).     3. Calculating the power of a specific test of $H_0$.

• Variables for Power:     * $\alpha$ (two-sided alpha).     * $\Delta$ (Difference worth detecting) = $\mu_a - \mu_0$.     * $n$ (sample size).     * $\sigma$ (standard deviation).     * $\Phi(z)$ (cumulative probability of Standard Normal z-score).

• SIDS Power Example:     * Let $\alpha = .05$, which gives $z_{1 - .05/2} = 1.96$.     * Test $H_0: \mu = 3300$ vs. $H_a: \mu = 3000$.     * $\Delta = 3300 - 3000 = 300$.     * With $n = 10$ and $\sigma = 720$, $\Phi(-0.64) = .2611$.     * Power identified as approximately $0.258$.

Practical Application: DNA Extraction Grinding Step

• Scenario: Standard kit extracts an average of $5\mu g$ DNA. A scientist adds a grinding step for $10$ samples to increase efficiency.

• Data set (DNA in $\mu g$): $10, 8, 8, 7, 6, 4, 5, 9, 12, 4$.

• Hypotheses:     * $H_0: \mu = \mu_0$ (where $\mu_0 = 5$).     * $H_a: \mu > \mu_0$.

• Calculated Statistics:     * $\bar{x} = 7.33$.     * $s = 2.626$.     * $n = 10; df = 9$.     * $SE_{\bar{x}} = \frac{2.626}{\sqrt{10}} = 0.83$.     * $t_{stat} = \frac{7.3 - 5}{0.83} = 2.7$.

• Decision: Calculated $t$ ($2.7$) is greater than the critical $t$ ($1.83$) at $\alpha = 0.05$.

• Conclusion: $H_0$ is rejected. The grinding step significantly improves extracted DNA quantities.

Inference for Paired Samples

• Definition: Two samples where each data point in one sample is uniquely matched to a data point in the other sample.

• Examples:     * "Pre-test/post-test" designs.     * Cross-over trials.     * Pair-matching.

• Analysis Approach: For paired data, calculations are performed on the within-pair difference variable, often denoted as "DELTA" ($\text{DELTA} = \text{Value 1} - \text{Value 2}$ or vice versa).

Case Study: Oat Bran and LDL Cholesterol

• Objective: To determine if oat bran reduces LDL cholesterol ($mmol/L$).

• Study Design: Cross-over trial with a washout period. Subjects use a CORNFLK diet for two weeks, then OATBRAN for two weeks.

• Data ($n = 12$ subjects):     * Subject 1: CORNFLK $4.61$, OATBRAN $3.84$, DELTA $0.77$.     * Subject 2: CORNFLK $6.42$, OATBRAN $5.57$, DELTA $0.85$.     * Subject 3: CORNFLK $5.40$, OATBRAN $5.85$, DELTA $-0.45$.     * Remaining Delta values: $-0.26, 0.30, 0.86, 0.60, 0.62, 0.31, 0.72, 0.09, 0.16$.

• Hypothesis Test:     * $H_0: \mu_d = 0$.     * $H_a: \mu_d > 0$ (one-sided) or $H_a: \mu_d \neq 0$ (two-sided).

• Statistics:     * $\bar{x}<em>d = 0.3808$.     * $s_d = 0.4335$.     * $n = 12; df = 11$.     * $t</em>{stat} = \frac{0.38083 - 0}{.4335 / \sqrt{12}} = 3.043$.

• P-value Determination (Table C):     * For $df = 11$, $|t_{stat}| = 3.043$:     * One-tailed: $.005 < P < .01$.     * Two-tailed: $.01 < P < .02$.

• Computer Output (StaTable/SPSS):     * Two-tailed Sig: $0.011$.     * Mean Difference: $0.38083$.     * $95\% \text{ CI for Difference}$: $(0.1054 \text{ to } 0.6563)$.

• Interpretation:     * Testing $H_0: \mu = 0$.     * Two-tailed $P = 0.011$, providing good reason to doubt $H_0$.     * The difference is significant at $\alpha = .05$, but not at $\alpha = .01$.