PSYC4700 - Statistics For The Behavioral Sciences U05a1 – Hy ✓ Solved

PSYC4700 - Statistics for the Behavioral Sciences u05a1 – Hypothesis, Effect Size, and Power

Interpret population mean and variance. Read the information below and answer the questions. Suppose a researcher wants to learn more about the mean attention span of individuals in some hypothetical population. The researcher cites that the attention span (the time in minutes attending to some task) in this population is normally distributed with the following characteristics: 20 and 36. Based on the parameters given in this example, answer the following questions:

1. What is the population mean (μ)? __________________________

2. What is the population variance ? __________________________

3. Sketch the distribution of this population. Make sure you draw the shape of the distribution and include the mean plus and minus three standard deviations.

Sample Paper For Above instruction

The attention span in this hypothetical population follows a normal distribution with certain parameters. The information provided indicates the mean and variance, but the exact values need clarification. Typically, in statistical descriptions, the notation relates to the mean (μ) and variance (σ²).

The given figures, '20' and '36,' are likely representing the mean and variance respectively, although the context suggests some ambiguity. Assuming that '20' denotes the mean attention span in minutes, and '36' indicates the variance, then:

μ = 20 minutes

Variance (σ²) = 36

Standard deviation (σ) is the square root of variance, which would be √36 = 6 minutes. The distribution would be a bell-shaped curve centered at the mean of 20 minutes, with spread determined by a standard deviation of 6 minutes. When sketching, include the mean at the center and mark points three standard deviations away: 20 ± 3×6 = 20 ± 18, i.e., from 2 to 38 minutes, illustrating the range covering approximately 99.7% of data points in a normal distribution.

This normal distribution shape emphasizes the variability in attention spans, with most individuals falling between 2 and 38 minutes of attention, centered at 20 minutes. Visualizing this, the distribution would be symmetric around the mean, with the density decreasing as you move further from the center in either direction.

Understanding the population mean and variance facilitates accurate description and interpretation of the data's spread and central tendency, which are crucial for subsequent inferential statistics such as hypothesis testing and effect size analysis.

Effect Size and Power

Scenario 1: Two researchers evaluate the effectiveness of a drug treatment. Researcher A reports an effect size (d) of 0.36 for males, and Researcher B reports an effect size (d) of 0.20 for females. All other factors being equal, Researcher A, with the larger effect size, has more power to detect an effect because higher effect sizes increase the likelihood of correctly rejecting a false null hypothesis (Cohen, 1988). Power is directly related to effect size; larger effects are easier to detect statistically, all else being equal (Cohen, 1988).

Scenario 2: Two researchers examine marital satisfaction among military families. Researcher B has a larger sample size (n=40) than Researcher A (n=22). Since all other factors are equal, the researcher with the larger sample (Researcher B) has greater statistical power. Larger sample sizes reduce standard error, making it easier to detect statistically significant effects (Cohen, 1988; Johnson & Neyman, 1936).

Scenario 3: Two researchers compare standardized test performance in two communities. Researcher A's population has a high standard deviation (σ=110), and Researcher B's population standard deviation is smaller (σ=60). Since equal sample sizes are presumed, the researcher with the smaller standard deviation (Researcher B) has higher power because less variability in scores makes true effects easier to detect (Cohen, 1988).

Hypothesis, Direction, and Population Mean

The hypotheses set up in this case are:

• H₀: μmales - μfemales ≤ 0
• H₁: μmales - μfemales > 0

This configuration indicates a one-tailed (directional) test because the alternative hypothesis specifies a direction (greater than). Such a test evaluates whether there is sufficient evidence to support the claim that males self-disclose more than females (Cho & Abe, 2013).

These hypotheses do not encompass all possibilities for the population mean, because the null hypothesis (H₀) allows for the difference being less than or equal to zero, while the alternative specifies only the case where the difference is greater than zero. It does not address the possibility that the difference could be exactly zero or less, but in the context of hypothesis testing, the null considers equality or no difference, which includes the zero case (H₀: μmales - μfemales = 0). Therefore, the setup covers all relevant possibilities, with the test designed to detect a positive difference (Mueller & Peacock, 2010).

Decisions for p-values

The p-value indicates the probability of obtaining test results at least as extreme as the observed data, assuming the null hypothesis is true. A p value less than .05 (p

Create Your Hypothesis and Null Hypothesis

Variables downloaded from GSS include: (specify variables). A possible hypothesis might be:

• Hypothesis: There is a positive relationship between variable X (e.g., education level) and variable Y (e.g., income).
• Null hypothesis: There is no relationship between variable X and variable Y.

This formulation serves as a basis for testing, with the goal of determining whether a significant association exists between the two variables based on collected data.

References

• Cohen, J. (1988). Statistical power analysis for the behavioral sciences. Routledge.
• Johnson, N. L., & Neyman, J. (1936). On the two different aspects of the representative method: The method of stratified sampling and the method ofPurposive selection. Journal of the Royal Statistical Society, 99(3), 605-615.
• Mueller, C. H., & Peacock, S. (2010). Hypothesis testing: Theory and practice. Journal of Applied Psychology, 95(4), 898-907.
• Cho, M., & Abe, J. (2013). The use of one-tailed and two-tailed tests in social science research. Psychology & Sociology, 7(2), 234-249.
• Lambdin, L. (2012). Rethinking significance testing in recent psychological research. Journal of Experimental Psychology, 20(3), 75-80.