Statistics Answer Note #22

Question 1 Assume we have a dataset that includes 60 observations surrounding two variables of interest: (1) Soybean yields in bushels per acre (bu/acre) and (2) fertilizer treatment. Variable (1) is quantitative, while variable (2) is categorical; assume that there were four different fertilizer treatments tested. Assume also that the number of observations of each fertilizer treatment was the same for each group (i.e., 15 observations of each fertilizer treatment were collected). What are the total degrees of freedom? Answer In order to answer this question, we first need to understand what degrees of freedom are. Degrees of freedom represent the number of independent pieces of information that can vary in a statistical calculation or analysis. In other words, it is the number of values that are free to vary without any restrictions in a statistical study. In this scenario, we have 60 observations and two variables of interest - soybean yields in bushels per acre (bu/acre) and fertilizer treatment. The first variable, soybean yields, is quantitative, meaning it is a numerical variable that can be measured and compared using mathematical operations. The second variable, fertilizer treatment, is categorical, meaning it is divided into categories or groups. According to the scenario, there were four different fertilizer treatments tested with 15 observations for each group. This means that there are 15 observations for each of the four categories of fertilizer treatment (A, B, C, D). Therefore, the total number of observations that were used in the study is (4 groups x 15 observations per group =) 60. Now, in order to calculate the degrees of freedom, we need to subtract the total number of categories (groups) from the total number of observations. In this case, we have four categories of fertilizer treatment, so the degrees of freedom would be (60 observations - 4 categories =) 56. Therefore, in this study, the total degrees of freedom would be 56. Question 2 Assume we have a dataset that includes 60 observations surrounding two variables of interest: (1) Soybean yields in bushels per acre (bu/acre) and (2) fertilizer treatment. Variable (1) is quantitative, while variable (2) is categorical; assume that there were four different fertilizer treatments tested. Assume also that the number of observations of each fertilizer treatment was the same for each group (i.e., 15 observations of each fertilizer treatment were collected). Assume you conducted an ANOVA test for the dataset described above and calculated an F statistic of 6.74. Using a 5% significance level, what would be your response to the null hypothesis? Answer The question being posed asks about the appropriate response to the null hypothesis given the results of an ANOVA test on a dataset with quantitative and categorical variables. ANOVA, which stands for Analysis of Variance, is a statistical method used to compare the means between two or more groups. In this case, the two variables being compared are soybean yields and fertilizer treatment. The null hypothesis in this scenario is that there is no significant difference in soybean yield between the four different fertilizer treatments. This means that any differences in yield between the groups are due to chance and not the type of fertilizer used. To answer this question, we need to look at the results of the ANOVA test and determine if the calculated F statistic of 6.74 is statistically significant at a 5% significance level. To make this determination, we first need to calculate the p-value, which measures the probability of obtaining the observed results or more extreme results if the null hypothesis is true. In this case, the p-value is the probability of getting an F statistic of 6.74 or higher if there is truly no difference in soybean yields between the four fertilizer treatments. To calculate the p-value, we can use software or a table of critical F values. In this scenario, we are given that there were 15 observations for each fertilizer treatment, resulting in a total of 60 observations. Therefore, the degrees of freedom (df) for the numerator (numerator df) is 3 (4-1), and the degrees of freedom for the denominator (denominator df) is 56 (60-4). Using these values, we can look up the critical F value using a table or calculate it using software. Assuming a 5% significance level, the critical F value for a numerator df of 3 and a denominator df of 56 is 2.67. Since our calculated F statistic of 6.74 is higher than the critical F value, this means that the p-value is less than 0.05. This indicates that the observed results are unlikely to occur if the null hypothesis is true, and therefore, we can reject the null hypothesis. In other words, the ANOVA test has provided evidence that there is a significant difference in soybean yields between the four fertilizer treatments. However, we cannot determine which specific fertilizer treatment led to the significant difference as that would require further analysis and tests. Question 3 “The greater the variability ________ samples relative to the variability ________ samples, the stronger the evidence supporting a difference between sample means.” (Select either “Between” or “Within” for each blank.) Answer According to statistical theory, the greater the variability between samples relative to the variability within samples, the stronger the evidence supporting a difference between sample means. Here is a more in-depth explanation of this concept: When studying a population, researchers often take a sample of that population in order to make inferences about the entire population. However, there will always be natural variability within any sample, even if it is taken from the same population. This is known as within-sample variability. On the other hand, there may also be differences in the characteristics of different samples taken from the same population. This is known as between-sample variability. For example, if a researcher is studying the income levels of a certain demographic, they may take multiple samples from that population at different times. In some samples, the average income may be higher than in others due to factors such as economic changes or random chance. Now, when we compare the variability between samples to the variability within samples, we can get an idea of how much of the differences between sample means can be attributed to natural variation within the population, and how much can be attributed to actual differences in the characteristics being studied. In other words, the greater the between-sample variability relative to the within-sample variability, the stronger the evidence is for a true difference between sample means. To understand this concept better, let us consider an example where we are comparing the average heights of two different groups: group A and group B. If the difference between the average heights of group A and group B is much larger than the variability within each group, then this would provide strong evidence for a true difference between the two groups. On the other hand, if there is not much difference in the average heights of the two groups and the variability within each group is large, then this would not provide strong evidence for a difference between the two groups. One way to measure the between-sample variability relative to the within-sample variability is by calculating the F-statistic, which is the ratio between these two types of variability. A higher F-statistic indicates a stronger evidence for a difference between sample means. In conclusion, the greater the variability between samples relative to the variability within samples, the stronger the evidence for a true difference between sample means. This concept is important for researchers to keep in mind while conducting studies and interpreting their results, and understanding it can help in making more accurate conclusions. Question 4 What are the three basic assumptions in using ANOVA? What, if anything, can you do when these assumptions cannot be met? Please provide a thorough explanation. Answer Introduction: ANOVA (Analysis of Variance) is an important tool used in statistical analysis to compare the means of three or more groups. It is based on several assumptions that must be met in order for the results to be valid and reliable. In this answer, we will discuss the three basic assumptions in using ANOVA and what can be done if these assumptions cannot be met. Assumption 1: Independence The first assumption in ANOVA is that the observations within each group are independent of each other. This means that the value of one observation should not affect the value of another observation. For example, in a study comparing the exam scores of two classes, the scores of one student should not be influenced by the scores of another student. Violation of this assumption can occur when there is a correlation between the observations within a group. What can be done: If independence is violated, we can use a more robust statistical test, such as the KruskalWallis test, which does not assume independence. This test is used when the data does not meet the assumptions of ANOVA, such as when the data is ordinal or when the variances are not equal. Assumption 2: Equal Variances The second assumption is that the variances of the groups being compared are equal. This means that the variability within each group should be the same. Violation of this assumption can occur when one group has a much larger or smaller variance compared to the other groups. What can be done: If equal variances cannot be assumed, we can use a modified version of ANOVA called Welch's ANOVA. This test does not assume equal variances and can be used when the sample sizes are unequal or when the data is not normally distributed. Calculation: Welch's ANOVA test statistic is calculated as follows: F = [(SSB/(k-1))/(SSW/(n-k))] * [(N-k)/(N-1)] where SSB is the sum of squares between groups, SSW is the sum of squares within groups, k is the number of groups, and n is the total sample size. Assumption 3: Normality The last assumption is that the data follows a normal distribution within each group. Normality means that the values are symmetrically distributed around the mean, with most values falling close to the mean and fewer values in the tails. What can be done: If normality is violated, we can use a non-parametric test, such as the Kruskal-Wallis test, which does not require normality. Additionally, we can transform the data to make it more normally distributed. This can be done by taking the logarithm, square root, or cube root of the data. Conclusion: In conclusion, ANOVA is a powerful tool for comparing means of three or more groups but it is important to ensure that the three basic assumptions are met in order to obtain valid and reliable results. If these assumptions cannot be met, there are alternatives that can be used, such as non-parametric tests or modifying the ANOVA test. It is important to carefully consider and assess the data before choosing the appropriate statistical test. Question 5 Complete an ANOVA table with the following data. Sum of squares between groups, 34.297, within groups, 483.143, and total, 517.440. Degrees of freedom between groups is 3, within groups, 96, and total 99. Answer Completing an ANOVA (Analysis of Variance) table is an essential step in analyzing the differences between multiple groups. It helps us determine if there is a significant difference between the means of these groups and if this difference is not due to mere chance. In this case, we have been given the sum of squares between groups (SSb), within groups (SSw), and total (SST) along with their respective degrees of freedom (df). So, let's begin by filling out the ANOVA table. ANOVA Table: | Source of Variation | Sum of Squares (SS) | Degrees of Freedom (df) | Mean Square (MS) | F-statistic | | --------------------| -------------------| -----------------------| -----------------| ----------- | | Between Groups | 34.297 | 3 | 11.432 | F = 11.432 | | Within Groups | 483.143 | 96 | 5.038 | | | Total | 517.440 | 99 | | |