Finding Specified Data Values In Exercises 31–38, answer the q...

Question

Finding Specified Data Values In Exercises 31–38, answer the questions about the specified normal distribution.

Billboard Hot 100 The length (in seconds) of the 100 most popular songs during the week of May 5, 2021, can be approximated by a normal distribution, as shown in the figure. (Source: Spotify)

Graph showing the normal distribution of song lengths in seconds, with a mean of 187.9s and standard deviation of 32.8s.

b. What song length represents the 17th percentile?

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says the weights in kilograms of a certain breed of adult dogs are normally distributed. The mean weight is 30 kg with a standard deviation of 4 kg. What weight corresponds to the 17th percentile for this breed? And we're given 4 possible choices as our answers. For choice A, we have 26.2 kg. For choice B, we have 34.8 kg. For choice C, we have 28.1 kg, and for choice D, we have 30 kg. Now we need to find what weight corresponds to the 17th percentile for this breed. And recall that for the 17th percentile, that the cumulative probability of P is going to be equal to 0.17. And we can actually look up to see what Z value that this corresponds to using our tables. And when we look that up, we see that this corresponds to a Z value that is equal to approximately minus 0.95. And so now we need to use the Z score to figure out our weight. And for this, we'll need to recall our formula 4, or Z score. Recall that our Z score Z is going to be equal to. X minus mu, and that quantity is going to be divided by sigma. Where X here is going to be our weight. Mu is going to be the mean, and sigma is going to be our standard deviation. Now, we were given both our mean and standard deviation in the problem. So we'll substitute in all of our values into this formula. So for Z, we'll have minus 0.95. And this is going to be equal to the quantity of X, since that is the weight that we're looking for, minus our mean, which we're told in the problem, was 30 kg, so we'll have minus 30. In quantity, and dividing that quantity by our standard deviation, which we're told is 4 kg, so we'll divide this by 4. So now we need to solve this equation for X. So the first step is to multiply both sides of the equation by 4. Doing so, on the left hand side we'll have -3.8, and that is going to be equal to X minus 30. Now, to solve for X, we just need to add 30 to both sides of this equation. In doing so, we'll get that X is going to be equal to 26.2, and this actually has units of kilograms, since both are mean and standard deviation are given in kilograms. So this here is the answer that we're looking for for this problem. And if we compare that to our answer choices, we see that the correct answer choice corresponds to answer choice A. So I hope that this explanation has been useful, and I'll see you in the next video.

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Key Concepts

Normal Distribution

Percentiles

Z-scores

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