Topic 10

PSYC 220 - Psychological Statistics

Topic 10: Introduction to Hypothesis Testing

Review of Key Concepts

• Uncertainty: The foundational concept in statistics, representing a lack of surety about an outcome. Relevance in forecasting future events.

• Probability: It answers the question of likelihood, expressed numerically, indicating chances of occurrence.

• Distributions: Referring to the arrangement of values, frequently visualized using the normal distribution curve.

Understanding Distributions

• Distribution of X Scores:

  • Mean ($ar{X}$): The average of the scores in the dataset.

  • Standard Deviation ($ ext{SD}$): A measure of the amount of variation or dispersion in a set of values, calculated as
    $ext{SD} = rac{X - ext{Mean}}{n}$

  • Sampling Distribution: The distribution of sample means over repeated sampling.

• Z-Score: Measures how many standard deviations an element is from the mean. Calculated using:
  $Z = rac{X - ext{Mean}}{ ext{SD}}$

Example 1 – Weight of Rats

• Scenario: Normal distribution for birth weights of rats with

  • Mean ($ ext{μ}$) = 18 grams

  • Standard Deviation ($ ext{σ}$) = 4 grams

  • Sample size ($n$) = 16

• Question: What is the probability of selecting a sample of rats with a mean weight less than or equal to 15 grams?

Probability in Decision Making

• Small Probability Event:

  • Defined as an event where $p ext{ (probability)} ext{ is less than or equal to } 0.05$ (or 5%). This suggests the event is extremely unlikely to occur.

Two-Tailed Cutoff Z Scores

• Identifies extreme 5% of scores located in both tails of the distribution.

• Characteristics:

  • No prior information about the weights of the rats; considering both light and heavy rats as extremes.

  • The cutoff Z scores for significance:
    $Z = ext{±}1.96$

  • Distribution of probabilities: 2.5% in each tail.

Summary on Extreme Scenarios

• Observing a sample mean of weight $ext{≤} 15$ grams is considered

  • Statistically significant if P < 0.05 or |Z| > 1.96.

Logic of Hypothesis Testing

• Hypothesis Testing: A technique leveraging sample data to make inferences about the population.

• Considerations for a sample of rats where:

  • Population Mean ($ ext{μ}$) = 18 (assuming known $ext{σ} = 4$)

  • Sample Mean = 15

  • Statistical significance threshold: If the mean shows extreme deviation from expected ($P < 0.05$), reject the null hypothesis H0.

Example 2 – Weight of Rats with Prenatal Alcohol Exposure

1. Hypothesis: Evaluate the effect of prenatal alcohol on birth weight.

2. Population parameters remain:

   • Mean ($ ext{μ}$) = 18 grams

   • Standard Deviation ($ ext{σ}$) = 4 grams

3. Sample of rats = 16, mean ($= 15$). Assess the effect compared to population mean.

Understanding Hypotheses

• Hypothesis: Statement regarding population parameters, commonly predictive (e.g., $ext{μ} = 18$).

• Types of Hypotheses:

  • Null Hypothesis (H0): Typically states there is no significant effect.

  • Alternative Hypothesis (H1): Indicates evidence of a change or effect.

• The H0 and H1 hypotheses are mutually exclusive.

Goals in Hypothesis Testing

• Aim to reject the null hypothesis (H0). Examples:

  • Stating that a medication does not have an effect.

  • No difference between methods.

Criteria for Decision Making

• Alpha Level ($ ext{α}$): Probability value, often set at 0.05 to gauge statistical significance.

• Critical Regions: Define extreme regions under H0, utilizing Z-scores ($Z = ext{±}1.96$ for α = 0.05).

Assessment of Test Statistic

• Test Statistic: Computed from sample data to aid in hypothesis testing, e.g., the Z-test.

• The statistic indicates whether the observation falls within the critical region.

Hypothesis Testing Steps

1. State Hypotheses: Define H0 and H1 clearly.

2. Criteria for Decision Setting: Establish alpha level.

3. Sampling & Statistics Computation: Collect and analyze data.

4. Decision: Reject or fail to reject H0 based on comparison to critical region.

One-Tailed vs. Two-Tailed Tests

• Two-Tailed Tests: Standard method, assesses extremes in both directions.

• One-Tailed Tests: Specific predictions about direction (e.g., an increase or decrease).

  • Higher power for hypothesis rejection, yet risks oversight of potential effects in the untested direction.

Examples and Application Scenarios

• Example of Professor’s Study: Investigating the impact of extra homework on exam scores.

  • Null Hypothesis: No change in scores.

  • Alternative Hypothesis: Scores improve with homework.

Final Notes

• Best practice: Use two-tailed testing unless strong justification for one-tailed approach.

• Be cautious about selecting hypothesis tests based on prior data knowledge which can skew integrity of testing results.