There are 104 students in Professor Negroponte's statistics class, 49 math majors and 55 statistics majors.Sixty of the students live in the dorms and the rest live off campus. Twenty of the math majors live off-campus. At the beginning of cach day that Professor Negroponte's class meets, he randomly selects a member of the class to present the solution to a homework problem. Suppose the class meets 40 times during the semester and the selections are made with (3) replacement. Let X= the number of times a statistics major is selected to present a solution. (a) Is X a binomial random variable? Justify your answer. No. While the number of outcomes are binary.the trials are independent,and there is a fixed n, the probability of a success is not the same for cach trial. Yes. The number of outcomes are binary, the trials are independent, there is a fixed n.the probability of a success is the same for cach trial. Yes. The number of outcomes are binary, the trials are not independent,there is not a fixed n, the probability of a success is not the same for cach trial. No. While the number of outcomes are binary,there is a fixed n, and the probability of a success is the same for each trial, the trials are not independent. (b) Calculate the mean of X. square Round to two decimal places. (c) Calculate the standard deviation of X. square Round to two decimal places.

(a) Yes. The number of outcomes are binary, the trials are independent, there is a fixed n, the probability of a success is the same for each trial.<br /><br />(b) To calculate the mean of X, we need to find the probability of selecting a statistics major to present a solution.<br /><br />There are 55 statistics majors out of 104 students, so the probability of selecting a statistics major is 55/104 = 0.529.<br /><br />Since the selections are made with replacement, the probability remains the same for each trial.<br /><br />The mean of X can be calculated as the product of the number of trials (n) and the probability of success (p):<br /><br />Mean = n * p<br /><br />In this case, the class meets 40 times during the semester, so n = 40.<br /><br />Mean = 40 * 0.529 = 21.16<br /><br />Therefore, the mean of X is 21.16.<br /><br />(c) To calculate the standard deviation of X, we need to find the variance first.<br /><br />The variance can be calculated as the product of the number of trials (n), the probability of success (p), and the probability of failure (1-p):<br /><br />Variance = n * p * (1-p)<br /><br />In this case, the probability of failure is 1 - 0.529 = 0.471.<br /><br />Variance = 40 * 0.529 * 0.471 = 9.99<br /><br />The standard deviation is the square root of the variance:<br /><br />Standard Deviation Deviation = √9.99 = 3.16<br /><br />Therefore, the standard deviation of X is 3.16.