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

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

Overview

This study guide outlines the key topics and skills required for an upcoming statistics exam. The exam consists of a short in-class component and a Pearson online component, covering material from Chapters 5, 6, and 7 of the textbook. Students are expected to demonstrate proficiency in probability, normal distributions, confidence intervals, and hypothesis testing.

Chapter 5: Normal Distribution and Applications

Standard Normal Curve and Probabilities

The standard normal distribution is a continuous probability distribution with a mean of 0 and a standard deviation of 1. It is used to model many natural phenomena and is foundational in inferential statistics.

• Finding Areas Under the Standard Normal Curve: Use the standard normal table or calculator functions to find probabilities associated with specific z-scores.

• Calculating Probabilities: To find the probability that a standard normal variable falls within a certain range, use the cumulative distribution function (CDF).

• Finding Z-scores for Given Percentiles: The z-score corresponding to a percentile can be found using inverse normal functions.

• Applications: Use z-scores and probabilities to solve real-world problems involving normal distributions.

Formula:

$z = \frac{X - \mu}{\sigma}$

Example: Find the probability that a standard normal variable is less than 1.99:

$P(Z < 1.99) = 0.9767$

Chapter 6: Confidence Intervals for Population Means

Margins of Error and Confidence Intervals

Confidence intervals provide a range of values within which the population parameter is likely to fall. The margin of error quantifies the uncertainty in the estimate.

• Calculating Margins of Error: The margin of error depends on the standard deviation and sample size.

• Confidence Intervals for Population Means (Sigma Known): Use the z-distribution when the population standard deviation is known.

• Minimum Sample Size: Calculate the minimum sample size required to achieve a desired margin of error.

• Confidence Intervals for Population Means (Sigma Unknown): Use the t-distribution when the population standard deviation is unknown.

• Constructing and Interpreting Confidence Intervals: Interpret the interval in the context of the problem.

Formulas:

For known sigma:

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

For unknown sigma:

$CI = \bar{x} \pm t^* \frac{s}{\sqrt{n}}$

Example: Construct a 95% confidence interval for a mean with $\bar{x} = 12.1$, $\sigma = 1.34$, $n = 24$:

$CI = 12.1 \pm 2.925 \times \frac{1.34}{\sqrt{24}}$

Chapter 7: Hypothesis Testing

P-values and Test Statistics

Hypothesis testing is used to make inferences about population parameters based on sample data. The p-value measures the strength of evidence against the null hypothesis.

• Interpreting P-values: The p-value indicates the probability of observing the sample data, or something more extreme, if the null hypothesis is true.

• Tests for Population Means (Sigma Known and Unknown): Use z-tests when sigma is known and t-tests when sigma is unknown.

• Tests for Population Proportions: Use the z-test for proportions to compare observed and expected values.

• Interpreting Results: Use the p-value to decide whether to reject or fail to reject the null hypothesis.

Formulas:

Z-test for mean:

$z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$

T-test for mean:

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

Z-test for proportion:

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

Example: For a sample mean $\bar{x} = 12.1$, population mean $\mu_0 = 11$, $\sigma = 1.34$, $n = 24$:

$z = \frac{12.1 - 11}{1.34 / \sqrt{24}} = 2.925$

P-value: $p = 0.00172$

Calculator Usage for Statistical Functions

Using Calculator Functions

Statistical calculators can be used to compute probabilities, confidence intervals, and test statistics efficiently. It is important to document the function used, input variables, and output values.

• normalcdf(lower, upper, mean, sigma): Calculates the probability that a normal variable falls between the lower and upper bounds.

• 2-Test, Stats: Used for hypothesis testing for means and proportions.

• Show All Work: Always show the function, inputs, and outputs for full credit.

Example: To find $P(-1.99 < Z < 2.12)$, use:

normalcdf(-1.99, 2.12, 0, 1) = 0.9829973084

HTML Table: Summary of Key Statistical Tests

Test	When to Use	Formula
Z-test for mean	Population sigma known	$z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$
T-test for mean	Population sigma unknown	$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$
Z-test for proportion	Testing population proportion	$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

Additional info: The study guide emphasizes the importance of showing all work, including calculator steps, and interpreting results in context. Students should be prepared for story problems and randomized exam questions covering all possible topics from the chapters listed.