review

Chapter 10: Sampling Surveys

  • Sampling Rules:

    • Sample must be representative of the population.

    • Randomness enhances sample representativeness.

    • Sample size matters; larger sample sizes provide similar randomness levels.

  • Sampling Methods:

    • Simple Random Sampling: Basic method of selecting samples randomly.

    • Stratified Sampling: Dividing population into strata and sampling from each.

    • Cluster Sampling: Dividing the population into clusters and randomly selecting clusters.

    • Multistage Sampling: Combination of sampling methods.

    • Systematic Sampling: Selecting samples using a fixed interval (e.g., every kth individual).

Chapter 11: Observational Studies and Experiments

  • Observational Studies:

    • Retrospective Study: Looks backward in time to study outcomes.

    • Prospective Study: Follows subjects forward in time.

    • Cannot establish cause-and-effect relationships directly.

  • Experiments:

    • Require random assignment of participants into treatment groups.

    • Four Principles of Experiments:

      1. Control What You Can: Keep conditions the same except for treatments.

      2. Randomize: Use random assignment to control variability.

      3. Repeat: Test with multiple subjects and settings.

      4. Blocking Design: Optional principle for controlling variability.

  • Placebo Effect:

    • Control group receives placebo (e.g., fake pills).

    • Ensures unbiased results by disguising group assignments.

Chapter 12: From Randomness to Probability

  • Types of Probabilities:

    • Empirical Probability: Frequency of an event occurring in many trials.

    • Modeling Probability: Ratio of favorable outcomes to total outcomes in the sample space.

  • Law of Large Numbers:

    • States that the probability will settle around a constant value over a long run.

    • No Law of Averages: Each event is independent of previous events.

  • Key Probability Rules:

    • Probabilities must be between 0 and 1.

    • Sample space probability equals 1.

    • Use complement rule for 'not' events.

    • Addition rule for mutually exclusive events: P(A or B) = P(A) + P(B).

    • Multiplication rule for independent events: P(A and B) = P(A) * P(B).

Chapter 13: Sampling Distributions

  • Sampling Distribution:

    • Distribution of all possible sample means or proportions.

  • Central Limit Theorem:

    • If sample size is large enough, sampling distribution will be approximately normal regardless of the population's distribution.

    • Mean of the sampling distribution equals population mean (μ). Standard deviation equals population stddev (σ) divided by the square root of sample size (n).

Chapters 14-19: Confidence Intervals and Hypothesis Testing

  • Z Intervals and Tests:

    • One-Proportion Z Interval: Estimate for a single proportion.

    • Two-Proportion Z Interval: Estimate compared between two proportions.

  • T Intervals and Tests:

    • One Sample T Interval: For means of one group.

    • Paired T Interval: For dependent groups (e.g., before and after).

  • Chi-Square Tests:

    • Goodness of Fit Test (GOF): Tests if observed counts match expected counts.

    • Tests for Homogeneity & Independence: Compare distributions across different groups.

  • Choosing the Right Test:

    • Analyze data type (categorical or quantitative) and number of groups to select appropriate interval or test.


Examples of Sampling Methods:

  • Simple Random Sampling: Drawing names from a hat to select a sample of individuals.

  • Stratified Sampling: Dividing a population of students into groups based on their year (freshman, sophomore, etc.) and sampling from each group.

  • Cluster Sampling: Selecting entire classes as clusters to study, instead of individual students, to represent a school.

  • Multistage Sampling: Choosing school districts first, then selecting schools within those districts, and finally sampling students from those schools.

  • Systematic Sampling: Selecting every 10th person from a list of registered voters for a poll.

Examples of Types of Studies:

  • Retrospective Study: Examining past medical records of patients to see the long-term effects of a medication.

  • Prospective Study: Following a group of volunteers who are exercising to see how it affects their health over time.

Examples of Experiment Principles:

  1. Control What You Can: Keeping temperature constant during an experiment testing plant growth.

  2. Randomize: Assigning participants randomly to treatments in a clinical trial.

  3. Repeat: Conducting the same experiment multiple times to verify results.

  4. Blocking Design: Separating subjects into blocks such as age groups and then randomly assigning treatments within those blocks.


Step-by-Step Guide to Solving Hypothesis Testing:

  1. State the Hypotheses:

    • Null Hypothesis (H0): The statement being tested, usually representing no effect or no difference.

    • Alternative Hypothesis (H1 or Ha): The statement that indicates the presence of an effect or difference.

  2. Choose the Significance Level (α):

    • Common levels are 0.05, 0.01, and 0.10, which indicate the probability of rejecting the null hypothesis when it is actually true (Type I error).

  3. Select the Appropriate Test:

    • Based on data type (categorical or quantitative) and sample size, choose between tests such as Z-tests, T-tests, or Chi-square tests.

  4. Determine the Test Statistic:

    • Calculate the test statistic using sample data. This could be a Z-score, T-value, etc., depending on the test chosen.

  5. Find the Critical Value(s):

    • Use statistical tables or software to find the critical value(s) that correspond to the significance level and the selected test.

  6. Draw a Conclusion:

    • Compare the test statistic to the critical value(s):

      • If the test statistic falls in the critical region (beyond the critical value), reject the null hypothesis.

      • If the test statistic does not fall in the critical region, fail to reject the null hypothesis.

  7. Make a Decision:

    • State the conclusion in the context of the problem, indicating whether you have sufficient evidence to support the alternative hypothesis.

  8. Report the Results:

    • Include the test statistic, the critical value(s), the significance level, and your conclusion.


Step-by-Step Guide to Solving Sampling Distributions:

  1. Define the Population:

    • Identify the population from which samples will be drawn and define the parameter of interest (mean, proportion, etc.).

  2. Select the Sample Size (n):

    • Decide on the sample size. The larger the sample size, the better the properties of the sampling distribution (e.g., more normality).

  3. Determine the Sampling Method:

    • Choose a sampling method (Simple Random Sampling, Stratified Sampling, Cluster Sampling, etc.) to ensure the sample is representative of the population.

  4. Collect Sample Data:

    • Randomly select n individuals from the population and collect data relevant to your parameter of interest.

  5. Calculate the Sample Statistic:

    • Compute the statistic (mean, proportion, etc.) from the sample data. For example, calculate the sample mean (x̄) if studying means.

  6. Construct the Sampling Distribution:

    • If possible, repeatedly draw samples of the same size from the population, and calculate the sample statistic for each draw.

    • Plot the distribution of these sample statistics to visualize the sampling distribution.

  7. Apply the Central Limit Theorem:

    • If the sample size is sufficiently large (typically n ≥ 30), the sampling distribution of the mean will be approximately normally distributed, regardless of the population's distribution.

  8. Calculate the Mean and Standard Deviation of the Sampling Distribution:

    • The mean of the sampling distribution (μx̄) equals the population mean (μ).

    • The standard deviation of the sampling distribution (standard error, SE) is calculated as: SE = σ / √n, where σ is the population standard deviation.

  9. Use the Sampling Distribution for Inferences:

    • Utilize the properties of the sampling distribution to make statistical inferences, such as constructing confidence intervals or performing hypothesis tests based on the sample statistics.


To find the critical value (z*) for hypothesis testing or confidence intervals, follow these steps:

  1. Determine the Desired Confidence Level: Common confidence levels are 90%, 95%, and 99%.

  2. Convert the Confidence Level to Alpha (α):

    • Alpha is the total area in the tails of the normal distribution outside the confidence level.

    • For example, for a 95% confidence level, α = 1 - 0.95 = 0.05.

  3. Calculate the Tail Area:

    • Since z* corresponds to the area in one tail, divide α by 2 (for two-tailed tests).

    • For a 95% confidence level, the area in each tail would be 0.05 / 2 = 0.025.

  4. Use the Z-Table or Standard Normal Distribution:

    • Look up the z-value in a standard normal (Z) distribution table that corresponds to the cumulative area of 1 - (α / 2).

    • For a 95% confidence level, look for the cumulative area of 1 - 0.025 = 0.975.

  5. Read the Corresponding z-value:

    • From the Z-table, you will find that the z-value corresponding to 0.975 is approximately 1.96. Therefore, z* = 1.96 for a 95% confidence level.

  6. Use Statistical Software (Optional):

    • Alternatively, you can use statistical software or online calculators to find the critical z-value directly by inputting the desired confidence level.

By following these steps, you can find the critical z-value (z*) needed for your statistical analysis.

Step-by-Step Guide to Finding the P-Value

  1. State Your Hypotheses:

    • Null Hypothesis (H0): The hypothesis that there is no effect or difference.

    • Alternative Hypothesis (H1 or Ha): The hypothesis that indicates the presence of an effect or a difference.

  2. Choose the Significance Level (α):

    • Common levels are 0.05, 0.01, and 0.10, representing the probability of rejecting the null hypothesis when it is actually true.

  3. Select the Appropriate Statistical Test:

    • Identify the type of test to be used based on the data type (categorical or quantitative) and sample size (Z-test, T-test, Chi-square test, etc.).

  4. Calculate the Test Statistic:

    • Using the sample data, compute the test statistic (e.g., Z-score, T-value, etc.).

  5. Find the Distribution of the Test Statistic:

    • Determine the distribution that relates to your test statistic (normal, t-distribution, chi-square distribution, etc.).

  6. Determine the P-Value:

    • Using statistical software, calculators, or tables, find the p-value corresponding to the calculated test statistic.

    • For a one-tailed test, the p-value is the probability of obtaining a result more extreme than the test statistic.

    • For a two-tailed test, the p-value is doubled since it accounts for both tails of the distribution.

  7. Compare the P-Value to α:

    • If the p-value is less than or equal to the significance level (p-value ≤ α), reject the null hypothesis.

    • If the p-value is greater than α, fail to reject the null hypothesis.

  8. Report the Results:

    • State the p-value and provide a conclusion in the context of the hypothesis test (indicating whether there is enough evidence to support the alternative hypothesis).

Example scenario: Finding a p-value for a t-test:

  • Problem:

    You want to test if the average height of a group of students is significantly different from 5 feet (60 inches) with a sample mean of 62 inches, a sample standard deviation of 3 inches, and a sample size of 20.

  • Steps:

    • Press "Stat" then "Tests". 

    • Select "T-Test" 

    • Choose "Stats" as the input method 

    • Enter the following values:

      • μ₀ (hypothesized mean): 60 

      • x̄ (sample mean): 62 

      • Sx (sample standard deviation): 3 

      • n (sample size): 20 

      • Select the appropriate alternative hypothesis (e.g., "≠" for a two-tailed test) 

    • Press "Calculate" to see the p-value displayed on the screen

Step-by-step for finding GOF

  • Enter data:

    • Press "STAT" to access the list editor. 

    • Input your observed data values into list L1. 

    • Input your expected data values (based on your theoretical distribution) into list L2. 

  • Access the test:

    • Press "STAT" then right arrow to "TESTS". 

    • Select option "chi-square GOF-Test" (usually labeled as "D"). 

  • Set parameters:

    • For "Observed" select "L1". 

    • For "Expected" select "L2". 

    • Enter the correct degrees of freedom (number of categories minus 1). 

  • Calculate:

    • Press "Calculate" to view the chi-square statistic and p-value