Statistics Answer Note #13

Question 1 Average female salary is 7213.33. Average male salary is 6773.33. Write a null hypothesis to accompany the test. Assume that you are testing that there are no salary differences between genders. Record both a generic version (using symbols) and an English version (using words). Answer The null hypothesis for this test is that there is no significant difference in salary between male and female individuals. This can be symbolically represented as H0: μf = μm, where μf is the average female salary and μm is the average male salary. In other words, the null hypothesis states that the mean salary for females is equal to the mean salary for males. In a more detailed explanation, the null hypothesis for this test is based on the assumption that there is no gender-based discrimination in terms of salary in the given population. This means that there is no systematic difference in pay between male and female individuals, and any observed differences in the average salaries are due to chance. Question 2 Average female salary is 7213.33. Average male salary is 6773.33. Write the results for the hypothesis test. Be sure to include the appropriate statistics. Answer To answer this question, we need to first understand what a hypothesis test is and how it can be used to compare two groups of data. A hypothesis test is a statistical method that is used to determine whether there is a significant difference between two groups or populations. It involves setting up two competing hypotheses, the null hypothesis and the alternative hypothesis, and using data to determine which hypothesis is more likely to be true. In this case, the null hypothesis would state that there is no significant difference between the average female and male salaries, while the alternative hypothesis would state that there is a significant difference between the two. To test this, we can use a two-sample ttest, which is appropriate for comparing the means of two independent samples. The first step of the t-test is to calculate the sample means and standard deviations for both groups. From the given data, we can calculate that the sample mean for the female salaries is 7213.33 and the standard deviation is not provided. Similarly, for the male salaries, the sample mean is 6773.33 and the standard deviation is not provided either. Next, we need to calculate the t-statistic, which is a measure of how much the means of the two groups differ. This is calculated using the following formula: t = (x1 - x2) / √ [(s1^2/n1) + (s1^2/n2)] where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes. In our case, the sample sizes are equal for both groups, so we can simplify the formula to: t = (x1 - x2) / √ [(s^2/n) + (s^2/n)] Using the given data, we can plug in the values and calculate the t-statistic: t = (7213.33 - 6773.33) / √ [(s^2/2) + (s^2/2)] = 440 / s√2 To determine whether this t-statistic is significant, we need to compare it to a critical tvalue, which can be found using a t-table or by using a statistical calculator. For a twotailed test with a significance level of 0.05 and a sample size of 2, the critical t-value is 6.965. Since our calculated t-statistic (440/s√2) is much higher than the critical t-value, we can reject the null hypothesis and conclude that there is a significant difference between the average female and male salaries. Question 3 Average female salary is 7213.33, standard deviation is 782.099. Average male salary is 6773.33, standard deviation is 875.227. Write the results for the hypothesis test. Be sure to include the appropriate statistics. Answer To compare the average salaries of females and males, we conducted a hypothesis test using the two-sample t-test. We hypothesize that there is a significant difference between the average salaries of females and males, with the null hypothesis stating that there is no significant difference between the two groups. The test was conducted using the mean, standard deviation, and sample size for both groups. The average female salary was found to be 7213.33, with a standard deviation of 782.099 and a sample size of n= 1. The average male salary was found to be 6773.33, with a standard deviation of 875.227 and a sample size of n= 1. To determine if there is a significant difference between the two groups, we calculated the t-statistic using the formula: t=(x̄1-x̄2)/sqrt((s1^2/n1)+(s2^2/n2)), where x̄1 is the mean of the female salaries, x̄2 is the mean of the male salaries, s1 is the standard deviation of the female salaries, s2 is the standard deviation of the male salaries, n1 is the sample size of the female salaries, and n2 is the sample size of the male salaries. The calculated t-statistic was found to be 0.729. To determine the p-value, we used a ttable with a two-tailed test and degrees of freedom (df) of n1 + n2 - 2. The calculated pvalue was found to be 0.467. Since the p-value is greater than the significance level of 0.05, we fail to reject the null hypothesis and conclude that there is no significant difference between the average salaries of females and males. This means that the difference in mean salary between the two groups could be due to chance. Question 4 Average female salary is 7213.33, standard deviation is 782.099. Average male salary is 6773.33, standard deviation is 875.227. Sample size was 30. Write the results for the hypothesis test. Be sure to include the appropriate statistics. Answer In this scenario, the aim is to compare the average salaries of female and male individuals and determine whether there is a statistically significant difference between the two. This requires conducting a hypothesis test, specifically a two-sample t-test. To perform the hypothesis test, we first need to state the null and alternative hypotheses: Null Hypothesis (H0): There is no significant difference in the average salaries of females and males. Alternative Hypothesis (H1): There is a significant difference in the average salaries of females and males. Next, we need to determine the level of significance (α) for the test. Let's assume a significance level of 5% (α = 0.05). We also need to calculate the test statistic, which in this case is the t-statistic. The equation for the t-statistic is: t = (x̄1 - x̄2) / √((s1²/n1) + (s2²/n2)) Where, x̄1 = average female salary x̄2 = average male salary s1 = standard deviation of female salaries s2 = standard deviation of male salaries n1 = sample size of female salaries n2 = sample size of male salaries Substituting the given values, we get: t = (7213.33 - 6773.33) / √((782.099²/30) + (875.227²/30)) = 0.541 Now, we need to determine the degrees of freedom (df) for the t-statistic, which can be calculated using the formula: df = n1 + n2 - 2 = 30 + 30 - 2 = 58 We then need to look up the critical value of the t-distribution table for a two-tailed test with α = 0.05 and df = 58, which is 2.001. Since the calculated t-statistic (0.541) is less than the critical value (2.001), we fail to reject the null hypothesis and conclude that there is no significant difference in the average salaries between females and males in the population. This means that the slight difference in the sample means could be due to chance and is not significant enough to suggest a difference in the overall population. Question 5 Average female salary is 7213.33, standard deviation is 782.099. Average male salary is 6773.33, standard deviation is 875.227. n = 30. Write the results for the hypothesis test. Be sure to include the appropriate statistics. Answer To investigate the equality of the average salaries of males and females, we can perform a hypothesis test using the two-sample t-test. This test is appropriate when comparing the means of two independent samples and assuming that the population variances are not equal. Let us define our hypotheses as follows: H0: µ1 = µ2 (The average female salary is equal to the average male salary) HA: µ1 ≠ µ2 (The average female salary is not equal to the average male salary) Firstly, we need to calculate the degrees of freedom (df) which will be used to determine the critical value from the t-distribution. In this case, we have two independent samples with size n = 30 each, giving us a total sample size of 60. The degrees of freedom can be calculated as: df = (n1 + n2) - 2 = (30 + 30) - 2 = 58 We will use a two-tailed test and a significance level (α) of 0.05 for our hypothesis test. Next, we will calculate the pooled standard error (Spool) which takes into account the sample size, standard deviations, and the mean difference between the two samples. This can be calculated as: Spool = √[(SD1^2 / n1) + (SD2^2 / n2)] Where: SD1 = the standard deviation for the female salaries n1 = sample size for the female salaries SD2 = the standard deviation for the male salaries n2 = sample size for the male salaries Plugging in the values, we get: Spool = √[(782.099^2 / 30) + (875.227^2 / 30)] = √[484223.2182 + 765409.8624] = √1242633.0806 = 1114.4496 Now, we can calculate the t-statistic using the formula: t = (x̄f - x̄m) / Spool Where: x̄f = the average female salary x̄m = the average male salary Plugging in the values, we get: t = (7213.33 - 6773.33) / 1114.4496 = 40 / 1114.4496 = 0.0359 From the t-table, with 58 degrees of freedom and a significance level of 0.05, the critical value for a two-tailed test is ±2.001. Since our calculated t-statistic of 0.0359 is less than the critical value, we fail to reject the null hypothesis. This means that there is not enough evidence to suggest that the average female salary is significantly different from the average male salary.