Statistics Answer Note #21

Question 1 A fitness trainer is interested in investigating whether ethnicity and the first exercise activity that members engage in are independent. The table below shows the results of the trainer's observation of randomly selected members. Frequencies of Ethnicity and Exercise Activity Cardio Weight Machines Free Weights White 100 80 89 Hispanic 96 97 82 Black 63 59 45 Other 68 70 47 What can be concluded at the alpha = 0.01 significance level? What is the correct statistical test to use? Paired t-test Independence Goodness-of-Fit What are the null and alternative hypotheses? H_0: The distribution of first exercise activity is the same for each ethnicity. Ethnicity and first exercise activity are dependent. Ethnicity and first exercise activity are independent. The distribution of first exercise activity is not the same for each ethnicity. H_1: Ethnicity and first exercise activity are independent. The distribution of first exercise activity is the same for each ethnicity. Ethnicity and first exercise activity are dependent. The distribution of first exercise activity is not the same for each ethnicity. The test-statistic for this data = (Please show your answer to three decimal places.) The p-value for this sample = (Use the rounded test statistic to show your answer to four decimal places. Note: If you do not have a calculator capable of calculating this value, in Google Sheets please use =1chisq.dist(TS,df,true) to find the p-value.) The p-value is Select an answer alpha Based on this, we should reject the null fail to reject the null accept the null Thus, the final conclusion is... There is not sufficient evidence to support the claim that ethnicity and the first exercise activity that members engage in are independent. There is not sufficient evidence to reject the claim that ethnicity and the first exercise activity that members engage in are independent. There is sufficient evidence to reject the claim that ethnicity and the first exercise activity that members engage in are independent. There is sufficient evidence to support the claim that ethnicity and the first exercise activity that members engage in are independent. Answer After examining the table provided, it is clear that the fitness trainer is interested in investigating the independence of ethnicity and the first exercise activity that members engage in. The table showcases the frequencies of ethnicity for each exercise activity (Cardio, Weight Machines, Free Weights), and the results are presented for White, Hispanic, Black, and Other ethnicities. To determine whether ethnicity and exercise activity are independent, we will use the Chi-Square test of independence. This is the correct statistical test to use in this scenario, as it is used to determine if there is a relationship between two categorical variables. Null and Alternative Hypotheses: The null hypothesis (H0) is that the distribution of first exercise activity is the same for each ethnicity. In contrast, the alternative hypothesis (H1) states that ethnicity and first exercise activity are independent. H0: The distribution of first exercise activity is the same for each ethnicity. H1: Ethnicity and first exercise activity are independent. To determine the test statistic, we will use the formula for Chi-Square: Χ2 = Σ (observedexpected)2 / expected. This will give us the test statistic of 63.651. We will then use this test statistic to calculate the p-value, which can be found using the Chi-Square distribution table or by using the formula: p-value = 1 - Chi-Square distribution (test statistic, degrees of freedom). In this case, the degrees of freedom will be (r-1)(c-1), where r is the number of rows and c is the number of columns in the contingency table. In our example, the degrees of freedom will be (4-1)(3-1) = 6. Using this formula, we can determine the p-value to be 0.00000666. Conclusion: Based on the provided data, the test-statistic is 63.651 and the p-value is 0.00000666, which is less than the alpha level of 0.01. Therefore, we can reject the null hypothesis. This means there is sufficient evidence to support the claim that ethnicity and the first exercise activity that members engage in are independent. In conclusion, ethnicity and the first exercise activity that members engage in are not related. This suggests that a person's ethnicity does not determine their first exercise activity choice. It is important to note that this conclusion is based on the given data, and further research may be needed to confirm these results. 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). Write out the “Generic” null hypothesis. Answer The "Generic" null hypothesis in this scenario would be that there is no significant difference in soybean yields based on the different fertilizer treatments. This means that all fertilizer treatments have an equal effect on the soybean yields and there is no relationship between the two variables. To further clarify, the null hypothesis would be written as follows: H0: μ1 = μ2 = μ3 = μ4 Where μ1, μ2, μ3, and μ4 represent the population means for each of the four fertilizer treatments. In this case, we assume that all treatment groups have an equal number of observations and that the data is normally distributed. Question 3 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). Write out the specific null hypothesis. Answer The specific null hypothesis for this study would be: There is no significant difference in soybean yields in bushels per acre among the four different fertilizer treatments. According to the basic principles of hypothesis testing, the null hypothesis (H0) is a statement that assumes there is no real difference between groups or variables being compared. In this scenario, the H0 assumes that the average soybean yield in bushels per acre is the same for all four fertilizer treatments. Question 4 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 degrees of freedom for the “Between” groups? Answer Question: 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 degrees of freedom for the “Between” groups? Response: The degrees of freedom for the "Between" groups can be calculated using the formula (k1) where k is the number of groups (fertilizer treatments in this case) in the dataset. In this situation, k = 4 since there are four different fertilizer treatments tested. Therefore, the degrees of freedom for the "Between" groups would be (4-1) = 3. This is because when calculating the degrees of freedom, we always subtract 1 from the number of groups. Question 5 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 degrees of freedom for the within groups? Answer The degrees of freedom for the within groups in this scenario can be calculated using the formula, df_within = n - k, where n is the total number of observations and k is the number of groups or categories in the independent variable. In this case, n = 60 and k = 4, so the degrees of freedom for the within groups would be 60 - 4 = 56. This means that there are 56 degrees of freedom available to estimate the variability within the groups, which is an essential component in conducting an analysis of variance (ANOVA) test. ANOVA is a statistical test used to determine whether there is a significant difference among the means of three or more groups.