Chapter 18: Inferences About Means – Confidence Intervals and Hypothesis Testing

Inferences About Means

Introduction

In statistical inference, we use sample statistics to estimate population parameters. Since sample statistics vary from sample to sample, we use probability models to quantify this variability and make inferences about the population mean μ.

Sampling Distribution for the Mean

Key Concepts

• Population parameter: The true value in the population (e.g., mean μ).

• Sample statistic: The value calculated from the sample (e.g., sample mean ̄y).

• Standard deviation of statistic: Measures the spread of the sampling distribution.

• Standard error (SE): Estimates the standard deviation of a statistic using sample data.

Population parameter	Sample statistic	Standard deviation of statistic	Standard error of statistic
Population proportion p	Sample proportion ̂p	$SD(\hat{p}) = \sqrt{\frac{p(1-p)}{n}}$	$SE(\hat{p}) = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
Population mean μ	Sample mean ̄y	$SD(\bar{y}) = \frac{\sigma}{\sqrt{n}}$	$SE(\bar{y}) = \frac{s}{\sqrt{n}}$

σ is the population standard deviation, s is the sample standard deviation.

Confidence Intervals for the Population Mean

Constructing Confidence Intervals

• When σ is unknown, use $SE(\bar{y}) = \frac{s}{\sqrt{n}}$ and the t-model instead of the standard Normal model.

• A confidence interval for μ with confidence level C is:

$\bar{y} \pm t^*_C \frac{s}{\sqrt{n}}$

• $t^*_C$ is the critical value from the t-table with $n-1$ degrees of freedom.

• This is called the one-sample t-interval.

Properties of the t-Model

• Symmetric about the mean 0, unimodal, and bell-shaped.

• Has one parameter: degrees of freedom (df), given by $n-1$.

• Thicker tails than the standard Normal model, especially for small sample sizes.

• As sample size increases, the t-model approaches the standard Normal model.

Finding Critical Values

• Use the t-table to find $t^*_C$ for the desired confidence level and degrees of freedom.

• If the exact df is not listed, use the closest available value.

Assumptions and Conditions for Confidence Intervals

• The sample is randomly drawn from the population.

• Sampled values are independent (sample size is a small fraction of the population size).

• The nearly Normal condition:

  • If the underlying distribution is exactly or nearly Normal, or unimodal and symmetric, the t-model is justified even for small samples.

  • If the underlying distribution is non-Normal or unknown, a large sample is needed for the t-model to work well.

Interpreting Confidence Intervals

• Over many repeated random samples, a confidence level C means that C% of the constructed intervals will contain the true mean μ.

• Example: A 99% confidence interval of (5.88, 9.42) means we are 99% confident that μ is between 5.88 and 9.42.

Worked Example: Confidence Interval for a Mean

• A student times 25 bus rides, finds $\bar{y} = 37$ min, $s = 2.5$ min.

• Construct a 90% CI for the mean travel time:

$SE(\bar{y}) = \frac{2.5}{\sqrt{25}} = 0.5$

$t^*_{0.90, 24} \approx 1.711$

$37 \pm 1.711 \times 0.5 = (36.14, 37.86)$

• Interpretation: We are 90% confident that the true mean travel time is between 36.14 and 37.86 minutes.

Hypothesis Testing for the Population Mean

Formulating Hypotheses

• Null hypothesis: $H_0: \mu = \mu_0$

• Alternative hypotheses:

  • $H_A: \mu \neq \mu_0$ (two-sided)

  • $H_A: \mu > \mu_0$ (right-tailed)

  • $H_A: \mu < \mu_0$ (left-tailed)

Test Statistic

• Use the t-model and estimate $SE(\bar{y})$:

$t_{n-1} = \frac{\bar{y} - \mu_0}{s/\sqrt{n}}$

• This is called the one-sample t-test.

Finding the P-value

• For two-sided tests: P-value is the sum of areas in both tails beyond the observed t-statistic.

• For one-sided tests: P-value is the area in the appropriate tail.

Decision Rule

• Reject $H_0$ if the P-value is smaller than the significance level $\alpha$.

• Do not reject $H_0$ if the P-value is larger than or equal to $\alpha$.

Worked Example: Hypothesis Test for a Mean

• Test if the mean travel time is longer than 35 minutes ($H_0: \mu = 35$, $H_A: \mu > 35$) using $\bar{y} = 37$, $s = 2.5$, $n = 25$.

• Test statistic: $t = \frac{37 - 35}{2.5/\sqrt{25}} = 4$

• Find P-value using the t-table (df = 24): P-value < 0.001

• At $\alpha = 0.10$, since P-value < 0.10, reject $H_0$.

• Conclusion: There is sufficient evidence to suggest the true mean travel time is longer than 35 minutes.

Appendix: Theoretical Foundations

• If $y$ is Normal, $\bar{y} \sim N(\mu, \frac{\sigma}{\sqrt{n}})$ for any sample size.

• If $y$ is not Normal, for large $n$, $\bar{y}$ is approximately Normal by the Central Limit Theorem.

• When $\sigma$ is unknown, use $s$ and the t-model with $n-1$ degrees of freedom.

Practice Problems and Solutions

Critical Values and Margin of Error

• Find $t^*$ for given df and confidence level using the t-table.

• Compute $SE(\bar{y})$ and margin of error: $ME = t^* \times SE(\bar{y})$

• Construct confidence intervals and interpret results in context.

One-Sample t-Test Applications

• State hypotheses, compute test statistic, find P-value, and make a decision.

• Interpret results in the context of the problem.

Linking Confidence Intervals and Hypothesis Tests

• If a confidence interval does not contain the null value $\mu_0$, reject $H_0$ at the corresponding significance level.

Additional info: These notes synthesize textbook content, lecture slides, and worked examples to provide a comprehensive overview of inference for means, including both confidence intervals and hypothesis testing using the t-model.