PSYC STATS EXAM 2 NOTES

Chapter 6: Probability

Overview

• Probability establishes a connection between populations and samples.

• It predicts the type of samples likely to be obtained from a population.

• It is defined as a fraction or proportion of all possible outcomes.

Importance of Probability

• Calculates the likelihood of obtaining a specific sample from a given population.

  • Example: Sample where M = 10, s = 5 from a population with μ = 20 and σ = 2.5.

  • Probability allows us to determine if the sample likely came from the specified population.

• In sampling distributions, the same mean is assumed, but a smaller standard deviation leads to a more concentrated curve.

Normal Distribution and Probability

• A vertical line drawn through a normal distribution divides it into two sections: body and tail.

• The z-score specifies the exact location of the line within the distribution.

• Percentile rank can be determined based on the z-score, indicating how often that z-score occurs.

• Positive proportions indicate likelihood, while a negative z-score implies body on the right and tail on the left.

Key Concepts in Probability

• Probability (p): p = specified outcome / total outcome

• Random Sampling: Each observation has an equal chance of selection; probability remains constant across selections (usually requires sampling with replacement).

• The requirement for constant probability is often ignored if the sample is less than 5% of the population.

Using the Unit Normal Table

Steps to find probabilities

1. Transform the score into a z-score.

2. Look up the z-score in the table to find the corresponding probability.

Standard Error of the Mean

• The standard error is denoted as σM = σ/√n, which describes the average distance between sample mean (M) and population mean (μ).

• As the sample size (n) increases, M becomes closer to μ based on the Law of Large Numbers which states:

  • Larger sample sizes result in smaller σM.

Central Limit Theorem

• The distribution of sample means across random samples defines its characteristics, highly significant for determining if a sample belongs to a population.

• Expected value of M (mean of the distribution) is always equal to the population mean (μ).

• Standard Error of M accounts for the variability and is computed by the relevant formula.

Properties of the Distribution of Sample Means

1. The shape tends to normalize under larger sample sizes (≥ 30) or if the population is normal.

2. Z-test applicability with the sample means leads to significant result findings.

• Formula: z = (M - μ) / σM.

Hypothesis Testing

Definition

• A statistical method that uses sample data to evaluate a hypothesis regarding a population.

Hypothesis Types

• Null Hypothesis (H0): Predicts no effect from the independent variable on the dependent variable.

• Alternative Hypothesis (H1): Predicts a significant effect from the independent variable.

Steps in Hypothesis Testing

1. State the hypothesis for the population.

2. Define the criteria probability (alpha level) for deciding the validity of the null hypothesis (typically set at .05).

3. Obtain and test a sample from the population and compute the relevant test statistic.

4. Compare the test statistic to the hypothesis predictions:

   • If in the critical region, reject the null hypothesis; otherwise, fail to reject it.

Errors in Hypothesis Testing

• Type 1 Error: Incorrectly rejecting the null hypothesis when it is true (false positive).

• Type 2 Error: Failing to reject the null hypothesis when it is false (false negative).

Testing Directionality

• Directional Tests: Include predictions in hypothesis statements affecting where the critical region lies.

• Non-directional Tests: Assess both positive and negative outcomes by including critical regions in both tails.

Effect Size and Power

• Effect Size: Measures the magnitude of an effect independent of sample size (39;Cohen's d42; is the most common measure).

• Power of a test: The likelihood of correctly rejecting the null hypothesis when an effect exists relies on:

  • Effect size

  • Sample size

  • Alpha level settings

• Measurement of Power: Aiming for an 80% chance to detect an effect under varying alpha levels.

t-Tests: Introduction and Assumptions

Overview

• t-tests are used when the population standard deviation is unknown, requiring sample estimates.

Assumptions

1. Interval or ratio scale measurement.

2. Random sampling.

3. Homogeneity of variance among data groups.

4. Normal distribution of sampled populations.

5. Independence of observations.

Types of t-Tests

1. One Sample t-test: Tests a sample mean against a hypothesized population mean.

2. Independent Samples t-test: Compares means between two distinct samples.

3. Dependent Samples t-test: Assesses means from repeating measures on the same subjects.

Independent Measures t-Test Procedure

1. State the hypothesis. Non-directional: H0: μ1 = μ2; Directional: H0: μ1 ≤ μ2.

2. Locate the critical region, calculating degrees of freedom accurately.

3. Calculate the t-statistic on two-sample estimates.

4. Make the decision based on the test results.

Conclusion on Results

• Measure effect size with Cohen’s d or percentage variance (r²). Reference critical values related to sample means and the overall effect which should be noted along with findings.