The last three questions in the examples of the previous section are reverse calculations because the questions gave you a percentage of weeks and asked you to find a miles driven per week value.ĭistinguishing between these two types of calculations (Table 6.1) is a matter of deciding if (i) the value of the variable is given and the percentage (or area) is to be found or (ii) if the percentage (or area) is given and the value of the variable is to be found. These calculations are called reverse questions simply to contrast them with the previous forward calculations. The second type of calculation occurs when you are given a percentage and asked to find the value (or values) of the variable related to that percentage. The first three questions in the examples of the previous section are forward calculations because the questions gave you miles driven per week values and asked you to find a percentage of weeks. These questions are called forward questions. ![]() The first type of calculation is when you are given a value of the variable (X) and asked to find a percentage of individuals. It is critical to be able to distinguish between two main types of calculations made from normal distributions. Between 135 and 185 – This is simply the first part of the 68-95-99.7% Rule.What are the most common 68% of miles driven?.The question is looking for the “lowest 2.5%” so it is the lower tail that starts exactly two standard deviations below the mean (i.e., 160-2×25=110). 110 – Hopefully the 2.5% sounds familiar from above, which relates it to one tail “left over” from 95%.What are the miles driven for the lowest 2.5% of miles driven?.The question is looking for the “highest 16%” so it is the upper tail that starts exactly one standard deviation above the mean (i.e., 160+1×25=185). 185 – Hopefully the 16% sounds familiar from above, which relates it to one tail “left over” from 68%.What are the miles driven for the highest 16% of miles driven?.The following questions, though they sound different, can also be answered with the 68-95-99.7% Rule. To get the area between 135 and 210, subtract these combined areas from 100. 210 miles is exactly 2 standard deviations above the mean (i.e., 160+2×25=210) and thus has 2.5% of the area above. 81.5% – 135 miles is exactly 1 standard deviation below the mean (i.e., 160-1×25=135) and thus has 16% of the area below it.What percentage of weeks does the driver drive between 135 and 210 miles?.The area between 1 standard deviation above and below the mean is 68%, so there is 32% outside of those two points with 16% in each tail. 84% – 185 is exactly 1 standard deviation above the mean (i.e., 160+1×25).What percentage of weeks does the driver drive less than 185 miles?.We want only one side of this symmetric distribution, so 5% is split in half to get 2.5%. ![]() The area between 2 standard deviations above and below the mean is 95%, so there is 5% outside of those two points.
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