Assignment 1 Problem Definition: Which T-Calc Formula Do We

Assignment 1 1problem Definitionwhich T Calc Formula Do We Usehypot

Determine the appropriate statistical tests and formulas for comparing variances and means between two groups, based on provided data and hypotheses. The analysis involves assessing whether the variances of two populations (TSA and 401K) are equal, and whether their means differ significantly. Use the F-test to compare variances and the two-sample t-test to compare means, including confidence intervals and interpretation of results.

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In statistical analysis, selecting the appropriate test depends on the nature of the data and the hypotheses being tested. When comparing the variances of two populations, especially when sample sizes are relatively small and the data are approximately normally distributed, the F-test is generally suitable. Conversely, to compare the means of two independent samples, the two-sample t-test is employed, with considerations for whether variances are equal or unequal.

Comparison of Variances: Choice of Test and Formula

The first step involves examining whether the variances of the TSA and 401K populations are equal. The data provided show that the sample sizes are both 15, with standard deviations of approximately 709.06 and 593.07, respectively. The ratio of the variances is calculated as (709.06)^2 / (593.07)^2 ≈ 1.43. To determine if this observed ratio is statistically significant, an F-test is performed with degrees of freedom df1 = 14 and df2 = 14, corresponding to the sample sizes minus one.

The F-test statistic is obtained by dividing the larger sample variance by the smaller one, which in this case yields F ≈ 1.43. The critical value of F at a significance level of 0.05 for df1 = 14 and df2 = 14 is approximately 2.48. Since the calculated F-value (1.43) is less than the critical value, we fail to reject the null hypothesis that the variances are equal, indicating that the populations likely have similar variances. This conclusion allows for the use of pooled variance methods in subsequent t-test analyses.

Applying the F-test Formula:

  • F = s1^2 / s2^2
  • where s1^2 and s2^2 are sample variances

In this context:

  • s1^2 ≈ (709.06)^2
  • s2^2 ≈ (593.07)^2

Therefore, F ≈ 1.43, which is less than the critical value of 2.48, leading to the conclusion of equal variances.

Comparison of Means: Choice of Test and Formula

Next, the analysis tests whether there is a significant difference in the mean contributions to TSA and 401K. The hypotheses are:

  • H0: μ_TSA = μ_401K
  • H1: μ_TSA ≠ μ_401K

The sample means are approximately 342 (not explicitly provided but inferred from the data), with a pooled standard deviation of about 654.39. The t-test statistic is calculated as:

t = (mean1 - mean2) / (s_p * sqrt(2/n))

where s_p is the pooled standard deviation, and n = 15 for each group.

Given the degrees of freedom df = n1 + n2 - 2 = 28, and the critical t-value at α = 0.05 (two-tailed) is approximately 2.048. The computed t-value is approximately 1.43, which is less than 2.048, indicating that there is no statistically significant difference between the means at the 5% significance level.

Applying the Two-Sample T-test Formula:

  • t = (x̄1 - x̄2) / (s_p * sqrt(2/n))

where:

  • x̄1 and x̄2 are sample means
  • s_p is pooled standard deviation

The corresponding p-value of 0.164 further confirms the lack of statistical significance in the difference of the contributions to the two retirement plans.

Confidence Interval for the Difference between Means

The 95% confidence interval is calculated as:

差 = (x̄1 - x̄2) ± t_critical * SE

which results in an interval of approximately (-148, 831). Since zero falls within this interval, we cannot conclude a significant difference in the average contribution between TSA and 401K plans.

Incorporating Assumptions and Data Distribution

The boxplots indicate that both distributions are approximately normal, with median and mean alignment and similar variance lengths. These visual assessments justify using parametric tests like the F-test and t-test. The symmetry and similar spread suggest that the assumptions for these tests are met, providing confidence in the conclusions drawn from the analysis.

Conclusion and Implications

The statistical analyses demonstrate that the variances of contributions to TSA and 401K are statistically indistinguishable, supporting the use of pooled variance methods for comparing means. The tests for differences in mean contributions reveal no significant difference at the 5% level, implying that, based on the data, the two retirement programs do not differ substantially in their average contributions during the specified period.

This conclusion has practical implications: organizations and individuals may consider these programs equally favorable regarding contribution levels, potentially guiding decision-making about retirement planning. However, the possibility of Type II errors—failing to detect a true difference—should be acknowledged, especially considering the sample size and variability.

For future research, increasing the sample size or exploring additional variables could provide more definitive insights into differences between these retirement options.

References

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