Threshold for low percentile | Modeling data distributions | AP Statistics | Khan Academy

The distribution of average wait times in drive-thru restaurant lines in one town was approximately normal, with a mean of 185 seconds and a 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% for that town. What is the maximum average wait time for restaurants where Amelia likes to use the drive-thru? 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. 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 is 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.

We want to find the maximum average wait time for restaurants where Amelia likes to use the drive-through. Well, what are those restaurants? That's where the average wait time is in the bottom 10% 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. 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%. Well, another way to think about it is this is the largest wait time for which you are still in the bottom 10%. 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 0.10 being less than that z-score. 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.

Remember, we're looking for 10 percent, but we don't want to go beyond 10 percent. We want to be sure that that value is within the 10th percentile; that any higher will be out of the 10th percentile. So, let's see, when we have these really negative z's, so far it only gets 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 in the hundredths place: 1, 2, 3, 4, 5, 6, 7, 8, 9. So, let's remember those columns.

Let's see if we are at a z-score of negative 1.28. Remember this is the hundreds, zero, one, two, three, four, five, six, seven, eight. So this right over here is a z-score of negative 1.28, 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 1.29, and this does seem to be the highest z-score for which we are within the 10th percentile. So, negative 1.29 is our z-score.

Now, 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 1.29 standard deviations below the mean. The negative says we're going below the mean, so we could say minus 1.29 times the standard deviation. They tell us up here the standard deviation is 11 seconds. So, it's going to be 1.29 times 11, and this is going to be equal to 1.29 times 11 is equal to 14.19.

Then I'll make that negative and then add that to 185; plus 185 is equal to 170.81. Now they say round to the nearest whole second. There are 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 would go to 171, but just by doing that, you might have crossed the threshold. But in all likelihood, for this application, where someone is concerned about wait time at drive-thru 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.