module 13 introduction to normal probability distributions 8 of 19. earlier we stated that for all normal probability curves the area within one standard deviation of the mean will equal 68 or 0.68 what is the probability of values falling outside this interval this is the question we will investigate now we will start with an example about normal probability curves in general and we will provide some specific examples for example probability x is more than one standard deviation above the mean in a normal probability distribution what is the probability that an x value is more than one standard deviation above the mean well the probability is 16 or 0.16 and here is how we know this look at the normal probability curve right so we have 68 percent is in here and 16 is on the tails and the tails okay here's within one standard deviation we already know they're 68 percent there right and what's left over is we have to well we know the total area is one and we already know what's in the middle one standard deviation above and below right if you take one and subtract 1 minus 0.68 in your calculator 1 minus 0.68 you should get point 32. right 32 percent but notice that this point 32 is going to have to be divided by two symmetrical spots right so point 32 is the total of the two tails but since we have two we can divide that by two and we get point sixteen for each of the tails so in other words the area under the probability curve always equals one the central area associated with x values within one standard deviation of a mean is 0.68 therefore the area outside of this region equals 1 minus 0.68 which is 0.32 this area is associated with x values more than one standard deviation from the mean both above and below this is the area in the two tails we want the probability that x values are more than one standard deviation above the mean this is the area in the tail to the right which is half of 0.32 this is how we got point 16 because it's half of 0.32 so example probably birth weight is less than 100 ounces birth weights have a normal probability distribution with a mean of 120 ounces and a standard deviation of 20 ounces what is the probability that a randomly selected infant weighs less than 100 ounces so we have you know the answer is 16 and let's look at how you reason through this using normal probability curve with mean 120 ounces and standard deviation of 20. right so mean is right here in the middle and one standard deviation above means we add 20 so that's 140 ounces one standard deviation below is 120 minus the standard deviation of 20 which is 100 right so we know that the area in each tail because we know the area inside you know between one center deviation below up to one standard deviation above the mean we know that's 68 percent so we also know that in the tails we have 16 percent right so area in the two tails combined is 32 or 0.32 area in each of the tails is half of that point 16 right so we are looking at uh what's the probability of being less than 100 ounces right and that's here's a hundred so it's going to be to the left if you notice to the left of 100 we have 80 60 etc and all the values in between so we're interested in really looking at the area and here all right so this is the interval we want birth weight's less than 100 ounces and the probability is the corresponding area which is percent or point sixteen now example probability of birth weight is more than one hundred ounces well birth weights have a normal probability distribution like before with a mean of 120 ounces standard deviation of 20 ounces what is the probability that a randomly selected infant weighs more than 100 ounces right more than 100 ounces so our answer is actually going to be well more than 100 ounces is going to be if we look at where 100 ounces is right here right it's going to be everything to the right of a hundred ounces so we are actually going to add the area in the red right more than 100 includes all this area in red plus over here above one standard deviation and we know that the area in red is 68 or 0.68 and we know the area to the right of the red is point 16. so if we add those up we get point eighty-four or eighty-four percent and that's how we get the answer right area in the two tails combined right we know that we know what area in each tail is right and so we're looking at this area to the right of a hundred in here and we get it by adding up the area in the red plus the leftover tail to the right