Sampling Distributions, Confidence Intervals, and Hypothesis Testing: Study Guide

Chapter 13: Sampling Distributions and Confidence Intervals for Proportions

Normal Sampling Distribution for One Sample Proportion

Understanding when the sampling distribution of a sample proportion is approximately normal is essential for inference about population proportions.

• Condition 1: Randomness – The sample must be randomly selected or the result of a randomized experiment.

• Condition 2: Independence – Observations must be independent. This is generally satisfied if the sample size is less than 10% of the population.

• Condition 3: Success/Failure – Both the expected number of successes and failures must be at least 10: $np \geq 10$ and $n(1-p) \geq 10$.

Constructing the Sampling Distribution for One Sample Proportion

• The sampling distribution of the sample proportion $\hat{p}$ is approximately normal if the above conditions are met.

• Mean: $\mu_{\hat{p}} = p$

• Standard deviation (Standard Error): $\sigma_{\hat{p}} = \sqrt{\dfrac{p(1-p)}{n}}$

Constructing Confidence Intervals for a Population Proportion

• A confidence interval estimates the true population proportion $p$.

• Formula: $\hat{p} \pm z^* \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$

• $z^*$ is the critical value from the standard normal distribution for the desired confidence level (e.g., 1.96 for 95%).

Finding an Appropriate Sample Size for a Given Margin of Error

• To achieve a desired margin of error $E$ for a confidence interval, solve for $n$:

$n = \dfrac{z^{*2} p^* (1-p^*)}{E^2}$

• If no preliminary estimate for $p$ is available, use $p^* = 0.5$ for the most conservative (largest) sample size.

Chapter 14: Confidence Intervals for Means

Normal Sampling Distribution for One Sample Mean

The sampling distribution of the sample mean $\bar{x}$ is approximately normal if:

• The population is normal, or

• The sample size is large ($n \geq 30$) by the Central Limit Theorem.

Constructing the Sampling Distribution for One Sample Mean

• Mean: $\mu_{\bar{x}} = \mu$

• Standard deviation (Standard Error): $\sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}}$

Constructing Confidence Intervals for a Population Mean

• When population standard deviation $\sigma$ is unknown (common in practice), use the $t$-distribution:

$\bar{x} \pm t^* \dfrac{s}{\sqrt{n}}$

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

Margin of Error and Length of Confidence Interval

• Margin of Error (E): The maximum expected difference between the true population parameter and a sample estimate.

• Increasing the sample size decreases the margin of error.

• Higher confidence levels increase the margin of error.

Chapter 15: Testing Hypotheses

Null and Alternative Hypotheses for One-Sample z or t Test

• Null Hypothesis ($H_0$): A statement of no effect or no difference (e.g., $H_0: \mu = \mu_0$ or $H_0: p = p_0$).

• Alternative Hypothesis ($H_a$): The claim we seek evidence for (e.g., $H_a: \mu \neq \mu_0$, $H_a: \mu > \mu_0$, or $H_a: \mu < \mu_0$).

p-value and Level of Significance

• p-value: The probability, assuming $H_0$ is true, of obtaining a result as extreme or more extreme than the observed result.

• Level of Significance ($\alpha$): The threshold for rejecting $H_0$ (commonly 0.05).

• If $p$-value $\leq \alpha$, reject $H_0$; otherwise, fail to reject $H_0$.

Performing a Hypothesis Test for a Population Proportion or Mean

• Test Statistic for Proportion (z-test):

$z = \dfrac{\hat{p} - p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}$

• Test Statistic for Mean (t-test):

$t = \dfrac{\bar{x} - \mu_0}{s/\sqrt{n}}$

• Compare the test statistic to critical values or use the p-value approach.

Chapter 17: Comparing Groups

Confidence Intervals for the Difference in Two Population Parameters

• For two proportions:

$(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\dfrac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \dfrac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

• For two means (independent samples):

$(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}$

Hypothesis Tests for the Difference in Two Population Parameters

• Set up null and alternative hypotheses (e.g., $H_0: \mu_1 = \mu_2$ or $H_0: p_1 = p_2$).

• Calculate the appropriate test statistic (z or t) and compare to critical values or use the p-value.

Chapter 18: Paired Samples and Blocks

Determining if Data Represents Paired Data

• Paired data arise when two measurements are taken on the same subject or matched subjects (e.g., before-and-after studies).

• Each pair forms a single difference value for analysis.

Confidence Intervals Associated with Paired Sample Data

• Calculate the differences for each pair: $d_i = x_{i1} - x_{i2}$.

• Construct a confidence interval for the mean difference $\mu_d$:

$\bar{d} \pm t^* \dfrac{s_d}{\sqrt{n}}$

• $\bar{d}$ is the mean of the differences, $s_d$ is the standard deviation of the differences, $n$ is the number of pairs.

Performing a Paired Sample t-Test

• Null hypothesis: $H_0: \mu_d = 0$ (no difference).

• Test statistic:

$t = \dfrac{\bar{d}}{s_d/\sqrt{n}}$

• Compare to the $t$-distribution with $n-1$ degrees of freedom.

Test Format and Study Strategies

• Test Format: 11 Multiple Choice Questions (MCQs), 4 Free Response Questions (FRQs).

• Resources Allowed: Calculator and one handwritten 3x5 inch index card (no worked examples).

• How to Study:

  • Complete review problems on Blackboard.

  • Review class notes and previous homework/quiz problems.

  • Prepare your index card with key formulas and concepts.