Statistical Hypothesis Notes

Introduction to Statistical Hypotheses

  • Definition of Statistical Hypotheses: Statistical hypotheses transform research hypotheses into statistical statements that can be tested using data.

Steps in Formulating Statistical Hypotheses

  1. Start with the Alternative Hypothesis (H1)

    • Always begin with defining the alternative hypothesis because it expresses what researchers hope to discover, such as differences or variations between groups.

    • Example: The statistical alternative hypothesis could state, "The average anxiety levels of senior citizens are lower than those of other adults."

  2. Corresponding Null Hypothesis (H0)

    • Once the alternative hypothesis is established, identify the null hypothesis, which represents a statement of no effect or no difference.

    • Example: In relation to the previous point, the null hypothesis would state that average anxiety levels of senior citizens are greater than or equal to those of other adults.

    • The null hypothesis typically covers all options that are not included in the alternative hypothesis.

    • Important: The null hypothesis always includes an equal sign, while the alternative hypothesis never includes an equal sign.

Types of Hypotheses

  • Alternative Hypothesis Options:

    • Less than

    • Greater than

    • Not equal to

Definitions and Operations

  • Left Side of Hypothesis Equation: Represents the group of interest (us). Commonly represented as .

  • Right Side of Hypothesis Equation: Represents the comparison group, generally the rest of the population.

Practice Problems

  1. Problem 1: Anxiety Levels and Computer Use

    • Research Question: Do people who spend more than four hours on a computer daily have worse eyesight than those who spend less?

    • Alternative Hypothesis (H1):  < ext{(Average eyesight of people who use computer less than 4 hours)}

    • Null Hypothesis (H0):  ext{ ≥ (Average eyesight of people who use computer less than 4 hours)}

  2. Problem 2: Extroversion Levels

    • Research Question: Do dog people have different extroversion levels compared to cat people?

    • Alternative Hypothesis (H1): ext{Dog people}
      eq ext{Cat people}

    • Null Hypothesis (H0): ext{Dog people} = ext{Cat people}

  3. Problem 3: Turtle Color Preference

    • Research Question: Do turtles prefer the color orange more than other colors?

    • Alternative Hypothesis (H1): ext{Preference for orange (u)} > ext{Preference for other colors}

    • Null Hypothesis (H0): ext{Preference for orange (u)} ≤ ext{Preference for other colors}

Error Types in Hypothesis Testing

  1. Type I Error (False Positive)

    • Reference: Occurs when a true null hypothesis is rejected; stating a difference exists when it does not.

    • Example: Incorrectly diagnosing someone as having a condition they do not actually have.

    • Significance Level/A: In psychology, the common threshold is set at ext{0.05} (5%). It signifies an acceptable risk of making a Type I error.

  2. Type II Error (False Negative)

    • Reference: Occurs when a false null hypothesis fails to be rejected; stating no difference exists when one does.

    • Generally, a Type II error is considered less critical than Type I errors.

Setting Up the Hypothesis Test

  1. Step 1: Identify Hypotheses

    • Determine both the null and alternative hypotheses based on research inquiries.

  2. Step 2: Set Type I Error Rate (A)

    • For most experiments in psychology, this is set at ext{0.05} or 5%.

  3. Step 3: Select Inferential Statistics

    • Currently using a z-test to evaluate hypotheses.

  4. Step 4: Find Critical Cutoff Values

    • Cutoff values determine where to reject the null hypothesis. For one-tailed tests:

      • Greater than: 1.65

      • Less than: -1.65

    • For two-tailed tests:

      • Greater than: 1.96

      • Less than: -1.96

    • Significance levels are calculated based on prior set Type I error rates. The cutoff must encapsulate the critical level where a null hypothesis is rejected if the observed z-score is equal to or beyond this.

  5. Step 5: Collect Data and Calculate z-Score

    • Formula for z-score:
      z = rac{x̄ - μ}{SE}
      where x̄ is sample mean, μ is population mean, SE is the standard error.

    • To compute standard error:
      SE = rac{ ext{Standard Deviation}}{ ext{sqrt(n)}}

  6. Calculate p-value from z-score

    • P-value signifies the probability of observing a data point as extreme or more extreme than the calculated z-score under the null hypothesis.

  7. Decision Making

    • Compare p-value with A:

      • If p-value < 0.05, reject null hypothesis.

      • If p-value ≥ 0.05, fail to reject null hypothesis.

    • Additionally, compare the calculated z-score with critical value:

      • If z-score is more extreme than critical value, reject null hypothesis.

      • If z-score is less than critical value, fail to reject null hypothesis.

Conclusion

  • Formulating statistical hypotheses is integral to hypothesis testing in research. By following structured steps, researchers can systematically test their theories through carefully formulated null and alternative hypotheses, leading to informed conclusions based on data analysis.

  • Practice clarifies these concepts as seen through various example scenarios.