discussion reply

Reply from Marilyn Cunningham

Marilyn Beazer

BUSI 820 – Comparing Groups Part 1

July 1, 2026

Discussion Forum 7

D7.9.1 One-Sample t-Test (a) A one-sample t-test is used when a researcher wants to compare the mean of a single sample to a known or hypothesized population mean. This test determines whether the sample mean differs significantly from the specified value. It is appropriate when the dependent variable is continuous, observations are independent, and the data are approximately normally distributed (Morgan et al., 2019). (b) Using the HSB dataset, a one-sample t-test could be conducted to determine whether students’ average mathematics scores differ significantly from a hypothesized population mean of 50. D7.9.2 Independent Samples t-Test (a) The equality of variances is determined by Levene’s Test. If the significance value is greater than .05, equal variances are assumed. (b) Insert the exact t, df, and p values from Output 9.2. (c) The statistically significant t-tests are those with probability values less than .05. (d) Students enrolled in the academic track demonstrated significantly higher mean grades in high school than students enrolled in the vocational track. Similarly, students in the academic track scored higher on visualization measures. (e) The 95% confidence interval provides a range within which the true population mean difference is likely to fall. (f) Effect sizes indicate the practical significance of findings. D7.9.3 Mann–Whitney U Test (a) The results of Outputs 9.2 and 9.3 should be similar when assumptions of the t-test are met. (b) The Mann–Whitney U test is used when data violate assumptions required for an independent samples t-test. D7.9.4 Paired Samples t-Test (a) The paired-samples correlation measures the strength and direction of the relationship between mothers’ and fathers’ education levels. (b) The paired-samples t-test evaluates whether a significant mean difference exists between mothers’ and fathers’ education levels. (c) The correlation measures the relationship between the two variables, whereas the t-test evaluates differences in their means. (d) If r = .90 and t = 0, the two variables would be highly correlated while showing no average difference between them. (e) If r = 0 and t = 5.0, there would be no relationship between mothers’ and fathers’ education levels, but there would be a substantial difference between their average educational attainment. D7.9.5 Wilcoxon Signed-Rank Test (a) Outputs 9.4 and 9.5 address similar research questions concerning paired observations. (b) The Wilcoxon Signed-Rank Test is appropriate when comparing two related samples with ordinal data or when the distribution of difference scores does not meet normality assumptions. References Morgan, G. A., Leech, N. L., Gloeckner, G. W., & Barrett, K. C. (2019). SPSS for introductory statistics: Use and interpretation (6th ed.). Routledge.

Reply from Jerolyn Hargrove

D7.9.1 (a) One sample t-test is a parametric statistical test that compares the mean of one continuous variable to some known or hypothesized mean. This test can be used in certain circumstances. First the variable of interest should be measured on a continuous level, usually interval or ratio. Secondly, the information should be independent, meaning, the result for one participant shouldn't be influenced by the result for another participant. Third, the sample must be drawn from a population that has approximately normal distribution, although the test is quite robust to violations of this condition, for large enough samples (e.g., n ≥ 30) by the Central Limit Theorem. To test a null hypothesis: μ = μ₀, one sample t-test is used. The test could be used on a group of university students, for instance, to see how their average IQ compares to the known average IQ of the general population, which is 100.

(b)A possible real world application of one sample t test to the High School and Beyond data set would be to determine if the mean writing score of a sample of 200 high school students is different than a proposed mean. For example, it is possible to compare the mean writing score (write) to a standard score, for example, 50. This analysis would involve comparing the sample mean with the test value (50) and seeing whether or not there is any statistically significant difference. The application can be useful for the purpose of the test, and for the one-sample t-test to be easily seen in action, a comparison is made between the mean of the sample and a fixed constant.

D7.9.2 (a) The F-statistic and associated significance level (Sig.) from an independent samples t test (Table D7.9.2 (a) output should be compared with one another to check if the variances are the same. If the p-value of Levene's test is more than the commonly accepted α-value of 0.05, then the null hypothesis that variances are equal is not rejected and the assumption of homogeneity of variance is accepted. If the p value is less than .05, however, then the variances may be different and the results for the row labeled "Equal variances not assumed" are interpreted.

(b)The value of each of the dependent variables would be chosen from the t test value table (Output 9.2), the values for the degrees of freedom (df) and probability (p) would be computed. They are typically presented in typical alphabetical list format. For instance, if the results are described for the task "grades in high school", then it could be t(df) = [t-value], p = [significance level]; or t(df) = [t-value], p = [significance level] for a second task; or t(df) = [t-value], p = [significance level] for a third task. The degrees of freedom (df), t-value and p-value will be the results of the Levene's test of each of the lines from the appropriate row.

(c)For a t-test, a p value of (c) < 0.05 is regarded as statistically significant. The p value comes in at less than the pre-determined alpha value of 0.05 (or less than 5%). So, if the p-value is less than 0.05, then all the dependent variables would be considered as statistically significant t test.

(d)The outcomes of t-test will be explained by taking into consideration the difference between the academic track per grade in high school. If the test is statistically significant, then it would indicate that the mean grade of high school students in one academic track (e.g., vocational) is significantly different from the mean grade of the high school students in another track (e.g., general). Similarly, a meaningful result for visualisation would be if there was a meaningful difference between the mean scores of the two academic tracks. An interpretation would be along the lines of the direction of the difference (more or fewer points – see "Mean Difference" in the output) to indicate which group had the higher average score.

(e) A 95% Confidence interval (CI) is an interval of values, likely to contain the actual difference between the two population means. For 'visualization' the same would apply as above: if the interval does not contain 0 then there is a statistically significant difference and the interval represents the size of the difference and the accuracy of the estimate (e.g., a narrow interval means that the estimation of the difference between the populations is accurate).

(f) Effect sizes are measures of the size of any difference observed; such as Cohen's d or Eta squared (η²) and are independent of sample size. The use of convention benchmarks of small (≈ 0.20), medium (≈ 0.50) and large (≈ 0.80) effects would be incorporated into the commentary on the effect size. For example, moderate effect sizes for the difference in high school grades between the academic tracks would imply that the difference is not only statistically significant, but of practical significance as well.

D7.9.3 (a) When comparing Outputs 9.2 and 9.3, it is apparent that the parametric independent samples t test is also similar to a non-parametric test (Mann-Whitney U test) but with some differences. The output 9.3, Mann-Whitney U test, is a non-parametric test, which does not require the assumption of normality and homogeneity of variance, while Output 9.2, t-test, is a parametric test and assumes normality and homogeneity of variance. With larger sample sizes, the results of the two tests may be comparable, but may differ if there are significant outliers or if the data are skewed or do not follow the normal curve. The Mann-Whitney U test is more robust to non-normality than is the raw score version of this test.

(b) Mann-Whitney U test can be used when the assumptions of the independent sample t-test are not met. In particular, it is used for data with an ordinal dependent variable or when the data is not normally distributed and the sample size is not large enough to apply the Central Limit Theorem. It is also resistant to the presence of outliers which can unduly affect the mean and standard deviation of the t-test. This test is used when the scores are not on a continuous interval/ratio scale or have a non-normal distribution and the central tendency (usually the median) of two independent groups need to be compared.

D7.9.4 (a) For Output 9.4, the paired samples correlation (0.787) between mother's and father's education is the strength and direction of linear relationships among the two paired variables. This correlation compares the level of education of mothers with fathers in families. It is not a difference test, but rather a measure of association and is basically a test for the co variation of the two variables.

(b) The results of the paired samples t-test are analyzed based on the t statistic, the degrees of freedom and the p value. This tests measure the null hypothesis: The mean difference between paired observations (e.g. mother's education versus father's education) is zero. The null hypothesis is rejected and the conclusion is that there is a statistically significant mean difference between the two variables in the population if the p-value is less than the 0.05 level of significance. The “Mean Difference” reported in the output would refer to whether there is a significant difference between the educational attainment of mothers and fathers (e.g., whether the mean attainment of fathers or mothers was higher).

(c) Correlation is a type of information, and the paired samples t-test is another type of information. The correlation (r) is to measure the strength and direction of the association or relationship between two variables and is not related to means. The t-test, on the other hand, only looks at the difference between the averages of the two variables and whether the average score on one variable is different from the average score on the other variable. There is no absolute correlation value that has a higher probability of yielding a significant t-test result (if the means are not very different), nor is there a correlation value that has a lower likelihood of yielding a significant t-test result (if the means are different but not always that way).

(d)The very strong positive correlation (r = .90) and the t = 0 in case (d) would indicate that there is a very high correlation between the education of a mother and a father. If a t statistic was zero, however, then the average of the differences between the paired scores would be zero, which means that on average the scores at the two education levels are the same. This pattern would suggest a high correlation between the variables; however there is no systematic difference between the means of the two.

(e) If the correlation (r) was zero and the t-statistic was 5.0, it would mean there is no linear relationship between mother's and father's education. A t-statistic of 5.0, however, which would normally be statistically significant, suggests that the mean difference between the two variables is not zero. This would show that there was no consistent pattern (that is, the level of education of one parent does not predict the other's) but there is a strong systematic difference (for example, mothers in general are significantly more or less educated than fathers).

D7.9.5 (a) Comparing Output 9.4 with Output 9.5 would help to make sense of the difference between a paired samples t-test and Wilcoxon signed-rank test, both of which are non-parametric. The paired samples t-test (Output 9.4) is a parametric test which assumes that the differences between paired observations are normally distributed. The Wilcoxon test (Output 9.5) is the non-parametric version of the Wilcoxon test. Unlike the Wilcoxon test, the test does not assume the data to be normally distributed because the Wilcoxon test compares ranks of the absolute differences between pairs. Therefore, when the assumption of normality of the difference is not satisfied and/or the data are ordinal, then Output 9.5 is more appropriate, as it provides results based on the signed ranks.

(b) If there are identical conditions for the Mann-Whitney U test for independent samples, the Wilcoxon signed rank test is the non-parametric test for paired samples t test. It is used if the assumption of normality does not hold true for the distribution of the difference scores, if the data are ordinal, or if the sample size is small and the Central Limit Theorem cannot be applied. The test is not affected by the mean and standard deviation and is very resistant to outliers; it is based on the ranks of the differences. It is the right test to apply when the parametric assumptions of the paired samples t-test are not met when comparing two related samples.

References

Krzywinski, M., & Altman, N. (2013). Points of significance: Significance, P values and t-tests. Nature Methods, 10*11), 1041-1042.

Laerd Statistics. (n.d.). One-sample t-test. https://statistics.laerd.comLinks to an external site.

Pallant, J. (2020). SPSS survival manual: A step by step guide to data analysis using IBM SPSS (7th ed.). Open University Press.