Statistics Answer Note #12

Question 1 If you calculate Levene’s statistic to test for equality of variances between two independent samples at the 5% (.05) significance level, and the output states “Sig. = 0.01” (i.e., p<0.01), should you conclude that the sample variances are assumed to be equal? Please explain your answer. Answer Levene’s statistic is a test used to determine whether the variances between two independent samples are equal or not. It is an important test in statistics as it helps determine the suitability of various statistical techniques for a given dataset. In this scenario, the output shows a significance level of 0.01, which is lower than the predetermined 5% significance level. This would lead one to believe that the sample variances are not equal. However, it is important to note that the conclusion should not be solely based on the significance level, but rather on a comprehensive assessment of the results. According to the American Psychological Association (APA), when using Levene’s test, the null hypothesis is that the variances of the two samples are equal. Therefore, a statistical significance test is conducted, and if the p-value is less than the chosen alpha level (0.05 in this case), the null hypothesis is rejected, and it can be concluded that the variances are not equal. However, if the p-value is greater than the alpha level, the null hypothesis cannot be rejected, and it can be assumed that the sample variances are equal. Further, it is important to understand that Levene’s test is not robust and can be influenced by outliers and non-normality in the data. In such cases, researchers are advised to use alternative tests, such as the Brown-Forsythe or Welch tests, which are more robust in such situations. Question 2 Assume you are comparing two independent samples and are not convinced that your data is normally distributed or that the variances of both samples are equal. What statistical test should you use to test our hypothesis of equivalent means across the two groups? Please explain your answer. Answer When comparing two independent samples, the most commonly used test for testing the hypothesis of equivalent means is the t-test. However, this test relies on certain assumptions, such as normality and equal variances, to be met in order for the results to be valid. Therefore, if we are not convinced that these assumptions are met, the t-test may not be appropriate and an alternative test should be considered. One such alternative is the non-parametric Mann-Whitney U test. The t-test is a parametric test, meaning it relies on the assumption that the underlying population is normally distributed. This assumption is important because the t-test calculates the probability of observing the difference in means between the two groups under the assumption that the data follows a normal distribution. If this assumption is violated, then the results of the t-test may not be reliable. In addition, the t-test also assumes equal variances between the two groups. In cases where this assumption is violated, the t-test may either overestimate or underestimate the true difference between the two group means. On the other hand, the Mann-Whitney U test is a non-parametric test, meaning it does not require the data to follow a specific distribution. This test is based on the ranks of the data rather than the actual values, making it more robust to violations of normality and equal variances assumptions. It does not make any assumptions about the underlying distribution of the data, making it a suitable alternative when the t-test assumptions are not met. To conduct the Mann-Whitney U test, we first rank the observations from smallest to largest, combining the two samples into one ranked list. We then calculate the sum of the ranks for each group, and use this information to calculate the Mann-Whitney U statistic. The test statistic follows a known distribution, and we can use this to calculate the pvalue and determine whether there is a statistically significant difference between the two groups. Question 3 Assume you are performing a one-tailed t-test with a 0.05 (α) level of significance and 24 degrees of freedom. What is the t-critical value? Answer The t-critical value in a one-tailed t-test is the point at which the probability of getting a tvalue that is equal to or greater than the critical value is equal to the chosen level of significance. In other words, it is the point at which the null hypothesis can be rejected with a certain level of confidence. To calculate the t-critical value for a one-tailed t-test, we first need to determine the alpha level (α) and degrees of freedom (df). In this case, the alpha level is given as 0.05 and the degrees of freedom is 24. We can then use a t-table or t-distribution function to find the critical value. Using a t-table, we find the value for alpha level (0.05) and degrees of freedom (24) intersecting at the table's tail end. This gives us a t-critical value of 1.711, which is the value at which we can reject the null hypothesis with 95% confidence. Alternatively, a tdistribution function in software such as Excel can also be used to calculate the t-critical value. In summary, the t-critical value for a one-tailed t-test with a 0.05 (α) level of significance and 24 degrees of freedom is 1.711. Question 4 Assume you are performing a one-tailed t-test with a 0.05 (α) level of significance and 24 degrees of freedom. If the t-statistic is +2.54, should you reject the null hypothesis? Please explain your answer. Answer The question at hand involves the use of a one-tailed t-test with a 0.05 (α) level of significance and 24 degrees of freedom, and whether or not the null hypothesis should be rejected given a t-statistic of +2.54. As a Masters student with high marks, I will provide a comprehensive answer to this question with a step-by-step breakdown of the calculations involved. Firstly, it is important to define certain key concepts in order to understand the significance of the results of a t-test. A t-test is a statistical test used to determine whether there is a significant difference between the means of two groups. The null hypothesis is the statement that there is no significant difference between the two groups, and the alternative hypothesis is the statement that there is a significant difference between the two groups. The alpha level, or level of significance, is typically set at 0.05, meaning there is a 5% chance of obtaining a result as extreme as the observed results if the null hypothesis is true. Degrees of freedom, in this context, refers to the number of independent pieces of information that are available in the data set. Moving on to the calculations, the first step would involve calculating the critical value for the t-test. This can be done by referring to a t-distribution table or by using statistical software. For a one-tailed t-test with 24 degrees of freedom and a 0.05 (α) level of significance, the critical value would be +1.711. This means that any t-statistic greater than +1.711 would be considered statistically significant, and the null hypothesis would be rejected. Next, we can compare the critical value to the t-statistic of +2.54. Since +2.54 is greater than +1.711, we can conclude that the results are statistically significant and the null hypothesis should be rejected. This is because the probability of obtaining a t-statistic of +2.54 or higher is less than 5%, which falls below the 0.05 (α) level of significance. It is also important to calculate the p-value, which represents the probability of obtaining results as extreme as the observed results, assuming the null hypothesis is true. In this case, the p-value for a t-statistic of +2.54 and 24 degrees of freedom is 0.008. This means that if the null hypothesis is true, there is only an 0.8% chance of obtaining a t-statistic of +2.54 or higher. As this p-value is lower than the 0.05 (α) level of significance, we can reject the null hypothesis. In conclusion, with a one-tailed t-test, an alpha level of 0.05 and 24 degrees of freedom, the critical value is +1.711 and a t-statistic of +2.54 falls above this value, indicating that the results are statistically significant. Additionally, the p-value is 0.008, indicating that if the null hypothesis is true, there is a very low chance of obtaining such results. Therefore, we can reject the null hypothesis and accept the alternative hypothesis that there is a significant difference between the two groups. Question 5 The table below contains monthly salaries for female and male sales reps at 10 different companies. The 10 companies were randomly selected, and one female and one male were randomly chosen from each company. If you were to run a paired-sample t-test, what would be the degrees of freedom? Answer The degrees of freedom for a paired-sample t-test can be calculated using the formula df = n-1, where n is the number of paired observations. In this case, since there are 10 pairs of monthly salary data (one from each company), the degrees of freedom would be 10-1 = 9.