Module 4 Problem Set Due Date October 28, 2015, Max Points 1

Module 4 Problem Set due Dateoct 28, 2015 235959max Points10details

The problems assigned here are intended to give you contextual experience with the types of statistics you will encounter as you conduct your dissertation research. Completing the assigned problems will increase your comfort level with these tools. General Requirements: Use the following information to ensure successful completion of the assignment: · This assignment is self-scored. · Locate the Module 4 Problem Set document. · Solutions are available. · Access the PSY-845 folder on the DC Network at helpful information. Directions: 1. Complete the problems in the Module 4 Problem Set. 2. Check your solutions by comparing your answers to the Module 4 Problem Set Solutions document. 3. Submit to the instructor a statement indicating that you have completed this assignment.

Paper For Above instruction

This paper addresses the core statistical concepts and procedures involved in conducting t-tests and z-tests within psychological research contexts, aligning with the requirements of the Module 4 Problem Set. The discussion elaborates on appropriate test selection, calculation methods, interpretation of results, and the implications of findings, especially in relation to experimental designs and hypothesis testing in psychology.

Introduction

Understanding the appropriate application and interpretation of statistical tests such as the t-test and z-test is fundamental for psychological research. These tools enable researchers to determine whether observed differences in data are statistically significant, informing conclusions about hypotheses related to populations and samples. Accurate test selection hinges on known variables such as population parameters, sample size, and the nature of the research design, particularly regarding whether data are independent or related.

Appropriate Test Selection Based on Data Characteristics

The choice between a t-test and a z-test primarily depends on knowledge about the population parameters. When the population mean and variance are known, the z-test is appropriate because it relies on the standard normal distribution. Conversely, if these parameters are unknown—which is common in psychological research—the t-test is preferred because it uses an estimate of the population variance derived from the sample. This estimate incorporates the variability inherent in smaller samples, improving the accuracy of significance testing. Notably, when the sample size is less than 30, even with known population parameters, the t-test is generally advised because it accounts for additional sampling variability.

Understanding the t-Statistic and Its Components

The t-statistic is calculated using an estimated standard error, known as the standard error of the mean (s_M), which is derived from the sample standard deviation or variance. This estimate substitutes the unknown population standard deviation and is critical for calculating the t-value. The formula involves the difference between the sample mean and the hypothesized or known population mean, divided by the standard error. This process allows researchers to assess whether the sample mean significantly deviates from the population mean, considering the variability within the sample.

Designs and Data in Independent Samples t-Tests

For an independent samples t-test, the experimental design requires participants to be randomly assigned to only one condition, ensuring independence between samples. When comparing two conditions, such as different lighting environments, the researcher must collect data that provides each group's mean, standard deviation, and sample size. These parameters are essential for calculating the degrees of freedom, which influences the shape of the t-distribution used for significance testing. The degrees of freedom (df) are computed as the total number of observations minus two or based on each group, depending on the specific test version.

Interpreting the t-Distribution and Critical Values

Once the t-value is computed, it is compared against critical values found in statistical tables (e.g., Table B.2 in the textbook). These critical values vary according to the degrees of freedom, the significance level (alpha), and whether the hypothesis is directional (one-tailed) or non-directional (two-tailed). The distribution of possible t-values under the null hypothesis forms the reference distribution, and the position of the calculated t within this distribution determines the statistical significance of the results.

Hypotheses, Tails, and Significance Testing

Hypotheses in t-tests can be nondirectional or directional. A nondirectional hypothesis predicts that there will be a difference between the means without specifying the direction, leading to a two-tailed test. A directional hypothesis specifies which mean is expected to be greater, guiding a one-tailed test. Depending on the hypothesis, the alpha level (commonly 0.05) is divided between the tails of the t-distribution: either all in one tail for directional tests or split equally in both tails for nondirectional tests. The critical t-value corresponds to the point beyond which only a specified proportion (e.g., 5%) of the distribution's values occur under the null hypothesis.

Hypothesis Testing and Significance

To determine significance, the calculated t-value is compared to the critical value from the relevant table. If the absolute value of the t exceeds the critical value, the result is statistically significant, allowing rejection of the null hypothesis. For example, with df=25 and alpha=0.05, the critical t-value for a two-tailed test is approximately 2.06. A computed t-value of 2.35 exceeds this threshold, indicating a significant difference between groups.

Effect Size: Cohen’s d

Beyond significance testing, assessing the magnitude of effects is important for understanding practical implications. Cohen’s d measures the standardized difference between two means, calculated as the difference between the means divided by the pooled standard deviation. Cohen’s d values around 0.2 are considered small, around 0.5 medium, and 0.8 or above large. In the context of comparing lighting conditions and honesty behaviors, a Cohen’s d of approximately 1.69 or 2.24 indicates a very large effect, signifying a substantial difference attributable to lighting conditions.

Case Example: Lighting and Dishonest Behavior

The study examining the impact of lighting on dishonesty involved two groups: well-lit and dimly lit rooms. After collecting data on the number of puzzles solved, the researchers conducted a t-test:

- Sample sizes were 9 per group.

- Means were 7.55 (well-lit) and 11.33 (dimly lit).

- Variances were calculated from sums of squares (SS), with SS of 42.22 and 38, respectively.

- The pooled variance and standard error were computed, leading to a t-value of approximately 3.57.

Since the critical t-value for df=16 at alpha=0.01 (two-tailed) is about 2.92, the observed t of 3.57 is statistically significant. The Cohen’s d was huge (~2.24), indicating a large treatment effect, meaning lighting significantly influences dishonest behaviors.

Conclusion

Statistical tests such as the t-test and z-test are essential tools in psychological research for evaluating hypotheses about population means based on sample data. Proper test selection depends on the knowledge of population parameters and study design. Correct calculation and interpretation of the t-value, degrees of freedom, and critical values enable researchers to make evidence-based decisions. Moreover, effect size measures like Cohen’s d provide insight into the practical significance of findings, strengthening the overall interpretation of results in psychological studies related to behavior, cognition, and experimental treatments.

References

• Gravetter, F. J., & Wallnau, L. B. (2017). Statistics for the Behavioral Sciences (10th ed.). Cengage Learning.
• Field, A. (2013). Discovering Statistics Using R. Sage Publications.
• Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum Associates.
• Keppel, G., & Wickens, T. D. (2004). Design and Analysis: A Researcher’s Handbook (4th ed.). Pearson.
• Howell, D. C. (2012). Statistical Methods for Psychology (8th ed.). Cengage Learning.
• Tabachnick, B. G., & Fidell, L. S. (2013). Using Multivariate Statistics (6th ed.). Pearson.
• Warner, R. M. (2012). Applied Statistics: From Bivariate Through Multivariate Techniques. SAGE Publications.
• Field, A., Miles, J., & Field, Z. (2012). Discovering Statistics Using R. Sage Publications.
• Zhong, C. B., Bohns, V., & Gino, F. (2010). Good Lamps Are Easy to Find: When Darkness Promotes Dishonesty. Psychological Science, 21(8), 1179–1181.
• Roth, P. L., et al. (2016). How to Conduct and Interpret T-Tests: Step-by-Step. Journal of Applied Psychology, 101(4), 583–599.