The instructor announced to his class that the recent exam scores were bell shaped so we can draw our bell shape and as soon as we hear that bell shape, we should be thinking empirical rule and the empirical rule says that in the center of the bell shape will be your mean And within one standard deviation of the mean. So that would be one standard deviation to the right and to the left. It will account for approximately 68% of the data. Within two standard deviations of the mean, so that would be to above or two below. That will account for 95% of the data, and within three standard deviations of the mean Will be approximately 99 7% of the data. Now, since the bell shaped curve is symmetric, then On each side of the mean will be congruent parts, mirror image, but congruent parts, so therefore the 68% can be divvied up into 34% to the left and 34% to the right. Or as a proportion That would be .34 And between two standard deviations away, there are four parts. Two of those parts are already defined to account for 68%. So that means there's 27% left over to be split, evenly making the section between two standard deviations Below the mean and one standard deviation below the mean to be 13,5 And the other part to be 135%. And as a proportion that would be .135. Since the 99.7% section is broken into six parts and four of them are already defined. They total up to 95%,, Meaning there is four 7% left to be split evenly. So that would make 2.35% in each section And as a decimal that would be .0235. So let's use this information to solve the problem that we have. So on this particular test we had a mean of 75 And a standard deviation of five. So we're going to draw our bell curve With that information so we're gonna play 75 in the center. We're going to Add a standard deviation will be at 80. If we added two standard deviations, we'd be at 85 And if we add three standard deviations we would be at 90 on the other side. If we subtract A standard deviation will be at 70. Subtract two. Standard deviations And subtract three standard deviations. So now we can also put in our proportions This would be 3, 4 and 34 Accounting for 68% of the data. This would be .135 And .135 accounting for mhm 95% of the data. And we'd have .0235 in each of these and three standard deviations away from the mean Would account for 99 7% of our data. So now we're ready to answer our questions in part A it's asking you what is the median of this data? Well, in a bell shaped curve, the mean, the median and the mod are all equal, And we were already told that the mean was 75, Therefore the median would have to be 75. In part B, we need to approximate How many students are, what proportion of students scored between a 70 and an 80. So if we look at our custom bell shaped curve between a 70 and an 80 would be 68% of the data Or as a proportion that would be 68 for part C. I recommend drawing the curve again. So we are trying to determine what proportion of students scored above 85. So if we draw that curve again He had 75 in the center and then we had 80 85 90 and 70 65 and 60. Then above an 80 you're sorry above an 85 Would account for these two parts And we know that between the 85 and the 90 was .0235 but we don't know the part above the 90. So the easiest approach to this is to start with the fact that A whole curve would be one. If we take away half the curve We're taking away five. Then if we take away the section between 75 and 80 then we are removing 0.34. And if we take away the section between the 80 and the 85 we are removing the .135. Therefore the proportion above an 85 is going to be point 0 to 5 or 2.5%. And then part D. What is the percentile rank Of an 85? So again I recommend drawing our picture, make it work for you, make it interactive. So when you're talking percentile rank, percentile means how many scored below you. So if you're scoring at the 30th percentile, that meant 30% did worse and 70% did better. So we need to know what percent did worse than an 85. So we're trying to figure out what percentile or what percent of the curve would be shaded. So from here on back is 50% in between 75 and 80 was 34 And between 80 and 85 was 13.5%. So when we add those parts together, 50% plus 34 Plus 13.5%. We find that 97 5% of the students scored lower than an 85. Therefore, the percentile rank of an 85 is the 95, sorry, 97.5 percentile.