Question

Graph showing the normal distribution of a sedan's braking distance, with mean 132 ft and standard deviation 4.53 ft.

What braking distance represents the first quartile?

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says the stopping distance of a motorcycle follows a normal distribution with a mean of 85 ft and a standard deviation of 3.2 ft. What stopping distance represents the third quartile Q3? And we're given 4 possible choices as our answers. For choice A, we have 87.2 ft. For choice B, we have 2.2 ft. For choice C, we have 84.1 ft, and for choice D, we have 65.9 ft. Now here we need to find the stopping distance that represents the 3rd quartile Q3. And we call that our 3rd quartile Q3. is the same as our 75th percentile. So that means that our third quartile represents a cumulative probability P of Z. That is equal to 0.75. So, we can use this cumulative probability to find our Z score, that corresponds to our 3rd quartile, and then use that Z score to figure out our stopping distance. So, when we look up in our tables, the Z score that corresponds to the cumulative probability of 0.75, we see that falls between two entries in our table. First entry is Z is equal to 0.67, and that has a cumulative probability P of Z. That is equal to. 0.7486. And the other entry is Z equal to. 0.68. Which has a probability P of Z, that is equal to 0.7517. So, in order to find the Z score that we're looking for, we'll need to use interpolation. So here the Z score that we're gonna be looking for, Z is going to be equal to, and here recall uh your formula for interpolation. So, our first term here is going to be the change in our Z values divided by the change in our probabilities from our table. So we'll have 0.68 minus 0.67, and that quantity is going to be divided by the quantity of 0.7517 minus 0.7486 in quantity, and then that fraction is going to get multiplied. By the difference of our probability that we have, that we're actually looking for with that of the lower bound. So in this case, our probability that we're looking for is going to be 0.75, and the smaller of the two probabilities that we have from the tables is going to be that that corresponds to Z equal to 0.67. So we'll have 0.75 minus 0.7486 in quantity. Then we'll have to add to that the Z score. Or the lower bound, which is going to be 0 point. 67. And when we Evaluate this expression, we get approximately equal to 0.6745. So now we need to use a Z score to find our stopping distance, and recall your relationship between the stopping distance and the Z score. That is going to be our stopping distance. X is going to be equal to, and here we call your formula, we're finding a value based upon the Z score, that's going to be equal to mu. Plus, Z multiplied by sigma. We're mu here is our mean and sigma is our standard deviation. And we were given both of those quantities and the problems. So that means our value for X is going to be equal to. Here we'll have our mean, which we're told is 85 ft in the problem, so we'll have 85. Plus, our Z score, which is 0.6745. Multiplied by our standard deviation sigma, which is 3.2 ft, and that was given in the problem as well. So, multiply that by 3.2. And when we evaluate this expression, we get that this is going to be approximately equal to. 87.2, and this actually has units of feet, since both our mean and standard deviation were given in feet. So this here is the answer that we're looking for for this problem. And if we look at our answer choices, we see that the correct answer choice corresponds to answer choice A. So I hope this explanation has been useful, and I'll see you in the next video.

What braking distance represents the first quartile?

Key Concepts

Normal Distribution

Quartiles

Z-Score

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