(Solved): 8. A very competitive scholarship is given to only the top 6% students who took a particular test. ...

8. A very competitive scholarship is given to only the top 6% students who took a particular test. The test has an average of 70 points with a standard deviation of 7, and the distribution of the points are symmetric about the average. A student got a 91 on the test, and she wants to know whether she guaranteed the scholarship. By using Chebyshev's theorem, help her to find out whether she is getting the scholarship for sure. Please use the following steps: (a) (points) Complete the statement of the Chebyshev's Theorem when u and o are the mean and the standard deviation of a random variable, respectively (b) (points) The percentage of scores better than 91 is P(X> 91), and she guaranties the scholarship if that is less than 6%, or in other words, she has to check P(X>91)< (c) (points) Find the number a such that P(X>91) = P(X < a) (d) (points) Find the numbers u and c such that 2P(X>91) = P(X>91) + P(X c)