Problem 3 In a college algebra course with 286 students in a...

Problem 3 In a college algebra course with 286 students in a lecture hall, the final exam scores have a mean of 68 and a standard deviation of 6. The grades on the exams are all whole numbers, and the grade pattern follows a normal curve. A. Sketch a normal curve, label the x-axis values at one, two and three standard deviations from the mean. B. Find the number of students who receive grades from one to two standard deviations above the mean. C. How many students earned a grade below 60%? D. Suppose the class only had 170 students. About how many students would earn a grade of 82 or more?

Transcript

00:01 Here we are told that the grades in the stats course followed a normal distribution with a mean of 75 and a standard deviation of 6.

00:10 So part a we were asked what proportion of the students had a final grade score of 72 or below.

00:17 So this is the probability that x is less than or equal to 72.

00:22 This graph represents our normal distribution for the grades.

00:26 We have a mean of 75 in the center.

00:29 Standard deviation of 6.

00:32 72 is approximately here.

00:35 The probability that x is less than or equal to 72 is equal to the area under the curve and to the left of 72.

00:43 So that corresponds to the area of this blue shaded region.

00:47 Now if we wish to use the standard normal distribution to solve this probability, or if we wish to use the standard normal table to solve it, we can standardize according to this formula.

01:01 So if we do that, it tells us that this is equal to the probability that z is less than or equal to minus .5.

01:11 So now we can look up z equals minus .5 in a standard normal table.

01:18 And that corresponds to a cumulative probability of .3085.

01:28 So the probability of a score less than equal to 72 is .3085.

01:35 And then for question b, we're asked for the proportion of students who earned a grade between 66 and 83.

01:47 Now this can be re -expressed as the probability that x is at most 83, minus the probability that x is less than 66.

01:59 And let's use excel to solve this one.

02:01 So let's solve the two terms in one step.

02:04 So if we go to excel, you want to start a computation with an equal sign...

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