Standard normal table for proportion below

a set of middle school students heights are normally distributed with a mean of 150 centimeters and a standard deviation of 20 centimeters darnell is a middle school student with a height of 161 point four centimeters what proportion of student heights are lower than Darnell's height so let's think about what they are asking so they're saying that heights are normally distributed so it would have a shape that looks something like that that's my hand-drawn version of it there's a mean of 150 centimeters so right over here that would be 150 centimeters they tell us that there's a standard deviation of 20 centimeters and Darnell has the height of 161 point 4 centimeters so Darnell is above the mean so let's say he is right over here and I'm not drawing it exactly but you get the idea that is 161 point 4 centimeters and we want to figure out what proportion of students Heights are lower than Darnell's height so we want to figure out what is the area under the normal curve right over here that will give us the proportion that are lower than Darnell's height and I'll give you a hint on how to do this we need to think about how many standard deviations above the mean is Darnell and we can do that because they tell us what the standard deviation is and we know the difference between Darnell's height and the mean height and then once we know how many standard deviations he is above the mean that's our z-score we can look at a Z table that tell us what proportion is less than that amount in a normal distribution so let's do that so I have my ti-84 emulator right over here and let's see Darnell is 161 point four centimeters 161 point four now the mean is 150 - 150 is equal to we could have done that in our head eleven point four centimeters now how many standard deviations is that above the mean well they tell us that a standard deviation in this case for this distribution is 20 centimeters so we'll take 11 point 4 divided by 20 so we will just take our previous answer so this just means our previous answer divided by 20 centimeters and that gets us zero point five seven so we can say that this is zero point five seven standard deviations deviations above the mean now why is that useful well you could take this z-score right over here and look at a Z table to figure out what proportion is less than zero point five seven standard deviations above the mean so let's get a Z table over here so what we're going to do is we're going to look up this z-score on this table and the way that you do it this first column each row tells us our z-score up until the tenths place and then each of these columns after that tell us which hundreds were in so zero point five seven the tenths place is right over here so we're going to be in this row and then our hundredths place is this seven so we'll look right over here so zero point five seven this tells us the proportion that is lower than zero point five seven standard deviations above the mean and so it is zero point seven one five seven or another way to think about it is if the heights are truly normally distributed seventy one point five seven percent of the students would have a height less than Darnell's but the answer to this question what proportion of students Heights are lower than Darnell's height well that would be zero point seven one five seven and they want our answer to four decimal places which is exactly what we have done