Date: 08/13/97 at 23:27:21 From: Amy Hunter Subject: Statistics A professor at a local college noted that the grades of her students were normally distributed with a mean of 74 and a standard deviation of 10. The professor has informed us that 6.3 percent of her students received an A, while only 2.5 percent of her students failed the course and received F's. What is the minimum score needed to make an A? What was the maximum score among those who received an F? And if there were 5 students who did not pass the course, how many students took the test? I can figure out the mean and the standard deviation, but am confused about what they want me to do, the kind of formula I should be using, and what lets me know that? Thanks. Amy Hunter
Date: 08/14/97 at 07:46:04 From: Doctor Anthony Subject: Re: Statistics x - m Use the formula z = ------- m = mean = 74, s = s.d. = 10 s The Normal tables give total probabilities up to a value of z. Sometimes you need the area beyond a value of z (upper tail area), so look up area corresponding to the value of z you have calculated and subtract this from 1. If you are given the probability, then you read the table backwards, i.e. you enter the area and find the corresponding value of z. (a) What is the minimum mark for an A? Here we shall be reading the table backwards. The upper tail area is 6.3 percent = .063, so look up in table an area equal to 1 - .063 = 0.937 and you will find this corresponds to z = 1.53 x - 74 So we have 1.53 = -------- and so x = 74 + 10 x 1.53 10 = 89.3 So a mark of 90 percent would be the minimum for a grade A. (b) What is the maximum mark of someone who failed? Again we read the table backwards, knowing that the tail area must be 2.5 percent = 0.025 Although this is the lower tail area, look up in the table an area 1 - .025 = 0.975 You will find this corresponds to a z value 1.96, but we are dealing with values below the mean (z = 0) and so z is negative, i.e. z = -1.96 x - 74 Therefore we put -1.96 = --------- so x = 74 - 10 x 1.96 10 = 54.4 So a mark of 54 percent would be the maximum of someone who failed. (c) If 5 students failed, how many took the test? Since 5 students corresponds to 2.5 percent of the total, simple proportion tells us that the total taking the test is 5 x 100 ------- = 200 students. 2.5 -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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