Threshold for low percentile

the distribution of average wait times in drive-through restaurant lines in one town was approximately normal with mean of 185 seconds and standard deviation of 11 seconds Amelia only likes to use the drive-through for restaurants where the average wait time is in the bottom 10 percent for that town what is the maximum average wait time for restaurants where Amelia liked to use the drive-through round to the nearest whole second like always if you feel like you can tackle this pause this video and try to do so I'm assuming you paused it now let's work through this together so let's think about what's going on they're telling us that the distribution of average wait times is approximately normal so let's get a visualization of a normal distribution and they tell us several things about this normal distribution they tell us that the mean is 185 seconds so that's 185 there the standard deviation is 11 seconds so for example this is going to be 11 more than the mean so this would be 196 seconds this would be another 11 each of these dotted lines or one standard deviation more so this would be 207 this would be 11 seconds less than the mean so this would be 174 and so on and so forth and we want to find the maximum average wait time for restaurants where Amelia likes to use the drive-thru well what are those restaurants that's where the average wait time is in the bottom 10 percent for that town so how do we think about it well there's going to be some value let me mark it off right over here in this red color so we're going to have some threshold value right over here where this is anything that level or lower is going to be in the bottom 10 percent well another way to think about it is this is the largest wait time for which you are still in the bottom 10 percent and so this area right over here is going to be 10% of the total or it's going to be 0.10 so the way we can tackle this is we can get up a Z table and figure out what z-score gives us a proportion of only 0.1 0 being less than that z-score and then using that z-score we can figure out this value the actual wait time so let's get our Z table out and since we know that this is below the mean the mean would be the 50th percentile we know we're going to have a negative z-score so I'm going to take out the part of the table that has the negative z scores on it and I remember we're looking for 10% but we don't want to go beyond 10% we want to be sure that that value is within the 10th percentile that any higher will be out of the 10 percentile so let's see when we have these really negative Z's so far it only gets it only doesn't even get to the first percentile yet so let's scroll down a little bit and let's remember as we do so that this is 0 and the hundredths place 1 2 3 4 5 6 7 8 9 let's remember those columns so let's see if we are at a z-score of negative one point two eight remember this is the hundreds of zero one two three four five six seven eight so this right over here is a z-score of negative one point two eight and that's a little bit crossing the 10th percentile but if we get a little bit more negative than that we are in the 10th percentile so this is negative one point two nine and this does seem to be the highest z-score for which we are within the 10th percentile so negative one point two nine is our z-score so this is Z equals negative one point two nine and if we want to figure out the actual value for that we would start with the mean which is 185 and then we would say well we want to go one point two nine standard deviations below the mean the negative says we're going below the mean so we could say minus one point two nine times the standard deviation and they tell us up here the standard deviation is 11 seconds so it's going to be one point two nine times 11 and this is going to be equal to one point two nine times 11 is equal to 14 point one nine and then I'll make that negative and then add that to 185 plus 185 is equal to 170 point eight one 170 point eight one now they say round to the nearest whole second there's a couple of ways to think about it if you really want to ensure that you're not going to cross the 10th percentile you might want to round to the nearest second that is below this threshold so you might say that this is approximately 170 seconds if you were to just round normally this will go to 171 but just by doing that you might have crossed threshold but in all likelihood for this application where someone is concerned about wait time it drive-through restaurants that difference in a second between 170 and 171 is not going to be mission-critical so to speak but you could say safely said 170 or 171 seconds will meet Amelia's needs