Statistics Answer Note #20

Question 1 We use a one-tailed t-test; our level of significance, α, is 0.01, and our degree of freedom is 20. If our computed t statistic is +1.89, should we reject or not reject our null hypothesis? Why? Answer According to the given information, we are using a one-tailed t-test with a level of significance (α) of 0.01 and a degree of freedom of 20. Our computed t-statistic is +1.89. To determine whether we should reject or not reject our null hypothesis, we need to compare our computed t-statistic with the critical value of t at α=0.01. The critical value of t at α = 0.01 and df = 20 can be found in a t-table. For this case, the critical value of t is +2.528. Since our computed t-statistic (+1.89) is lower than the critical value (+2.528), we do not have enough evidence to reject the null hypothesis. The p-value is another important factor in determining whether we should reject or not reject the null hypothesis. The p-value represents the probability of obtaining our sample result, assuming the null hypothesis is true. In this case, our computed t-statistic of +1.89 corresponds to a p-value of 0.0333, which is greater than the level of significance (α = 0.01). This means that there is a 3.33% chance of obtaining our sample result if the null hypothesis is true. Generally, a p-value greater than the level of significance indicates that we do not have enough evidence to reject the null hypothesis. Question 2 If a t-test yields a low probability, such as p < .05 or p < .01, what decision is usually made about the null hypothesis? Answer A p-value is the probability of obtaining a result as extreme or more extreme than the observed result, assuming that the null hypothesis is true. It is used to determine the significance of the results in statistical analysis. In general, a low p-value indicates that the observed result is unlikely to occur if the null hypothesis is true. Therefore, when a ttest yields a low probability, such as p < .05 or p < .01, it is usually concluded that the null hypothesis can be rejected. The decision made about the null hypothesis depends on the level of significance (alpha) chosen for the study, which is usually set at 0.05 or 0.01. If the calculated p-value is less than the chosen alpha, it suggests that the observed result is statistically significant and the null hypothesis can be rejected. On the other hand, if the p-value is greater than the chosen alpha, it indicates that the observed result is not statistically significant and the null hypothesis cannot be rejected. In research studies, the null hypothesis is a statement that there is no significant difference between the groups being compared. It is usually the default or starting assumption, and the researcher& Question 3 If you read the statement, “The null hypothesis was rejected (t = 2.810, df = 40, p < .01),” should you conclude that the difference between sample means was statistically significant? Answer Based on the given statement, it can be concluded that the difference between sample means was statistically significant. This is because the null hypothesis was rejected at a level of significance of p < .01, which indicates a very low probability that the results were due to chance. The t-value of 2.810 and degrees of freedom (df) of 40 also support this conclusion. The t-value is a measure of how far the sample means are from each other, and a higher tvalue suggests a greater difference between the two groups. In this case, a t-value of 2.810 is considered large and therefore, the difference between the sample means can be considered significant. Additionally, the degrees of freedom (df) is the number of scores that are free to vary in a calculation. In this case, the df is large enough (40) to support the accuracy of the t-value and validate the significance of the results. Moreover, the statement also mentions that the null hypothesis was rejected, which means that there is evidence to support the alternative hypothesis. In other words, the results of the study do not support the idea that the difference between sample means is due to chance. This further strengthens the conclusion that the difference between sample means was statistically significant. In terms of statistical calculations, a t-test was most likely used to determine the significance of the results. A t-test is a statistical test that is used to compare the means of two groups, and in this case, it was used to compare the sample means. The t-value and degrees of freedom are important components of the t-test, and they help in determining the level of significance. Additionally, the p-value plays a crucial role in determining the significance of results. In this case, with a p-value of less than .01, there is strong evidence to reject the null hypothesis and conclude that the difference between sample means is statistically significant. Question 4 You have a sample of 100 female salaries from 10 ag-related companies and 100 male salaries from 10 other ag-related companies. Assume the data in both samples are normally distributed. What statistical procedure should you use to test for salary differences between females and males? Why? Answer Based on the given information, the appropriate statistical procedure to test for salary differences between females and males is the independent samples t-test. This test is commonly used to determine if there is a significant difference between the means of two independent groups. In this case, the two groups are females and males from different agrelated companies. The independent samples t-test requires the assumption of normality, which is satisfied by the given information that the data in both samples are normally distributed. Additionally, the t-test also assumes that the variances of the two groups are equal, which can be tested using Levene's test for equality of variances. Question 5 You have a sample of 100 female salaries from 10 ag-related companies and 100 male salaries from 10 other ag-related companies. You are concerned that the data are not normally distributed. What statistical procedure should you use to test for salary differences between females and males? Why? Answer Based on the given scenario, a suitable statistical procedure to test for salary differences between females and males would be the 2-sample t-test, specifically the independent samples t-test. This test is used to compare the means of two independent groups, in this case the female and male salaries. The independent samples t-test assumes that the samples from each group are normally distributed, have equal variances, and the observations are independent within and between groups. However, as the data given may not meet the assumption of normality, an alternative non-parametric test such as the Mann-Whitney U test can also be considered. The main reason for choosing the 2-sample t-test is its ability to compare means of two groups, making it suitable for testing salary differences between females and males from different ag-related companies. Additionally, it is considered a more powerful test compared to the Mann-Whitney U test when the data does meet the normality assumption. To conduct the 2-sample t-test, the following steps can be followed: Step 1: State the null and alternative hypotheses. In this scenario, the null hypothesis would be that there is no difference in the mean salary between females and males, while the alternative hypothesis would be that there is a significant difference in the mean salary between females and males. Step 2: Choose an appropriate level of significance. This is the probability of rejecting the null hypothesis when it is actually true. Generally, a significance level of 5% is used, which corresponds to a confidence level of 95%. Step 3: Calculate the test statistic. For the 2-sample t-test, the test statistic is calculated as t = (x̄1 - x̄2) / √ [(s1^2/n1) + (s2^2/n2)], where x̄1 and x̄2 are the sample means of females and males, s1^2 and s2^2 are the sample variances, and n1 and n2 are the sample sizes. Step 4: Determine the critical value. This is based on the level of significance and the degrees of freedom. The degrees of freedom can be calculated as df = (n1 + n2) - 2. Step 5: Compare the test statistic with the critical value. If the test statistic is greater than the critical value, then the null hypothesis can be rejected, indicating a significant difference in the mean salaries between females and males.