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.

COVID-19 Response Surveyors asked respondents to rate ten key aspects of their government’s response to the COVID-19 pandemic, including preparedness, communication, and material aid. A pandemic response score that ranged from 0 to 100 was calculated. The mean score for U.S. respondents was 50.6 with a standard deviation of 29.0. (Source: PLOS One)

b. What score represents the 61st percentile?

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A standardized math test has scores that are normally distributed with a mean of 480 and a standard deviation of 60. What score represents the 42nd percentile? Awesome. So it appears for this particular prom, we're ultimately trying to determine what score represents the 42nd percentile. So now that we know that we're trying to find the score that represents the 42nd percentile, given the information that is provided to us by the prom itself, let's read off our multiple choice answers to see what our final answer might be. A is 468, B is 472. C is 495 and D is 453. Awesome. So our first step to solve this problem is we need to once again recall that we're trying to find the Z score for the 42nd percentile, and the area to the left is 0.42, as we should recall, and from our standard normal table, which our standard normal table would be located in our textbook and using that. To help us determine that the closest Z score is about -0.2. So once again, we need to use our standard normal table from our textbook to determine that the closest Z score to the area to the left of 0.42 will be about -0.20. Awesome. So our second step is we need to recall and use the Z score formula, which states that Z to Z score is equal to X minus mu, all divided by sigma. Where we need to note that the mean Which we're gonna do now as mu. So mu is the mean and mu, the mean is equal to 480, and our standard deviation, which is represented by sigma, is going to be equal to 60 as given to us by the problem itself. So our third step now is to simply plug in our known variables into our equation. So we know that Z in this case is going to be equal to -0.20 is equal to X minus. 480, all divided by 60, and in this case, we know all of our values but X, so that means we need to rearrange this expression to isolate and solve for X. So to do that, to begin, we need to multiply both sides by 60. And that is our 4th step. So when we multiply both sides by 60, we will get -0.20 multiplied by 60, is equal to X minus 480. Fantastic. So now if we add 480 to both sides, we will find that X will be equal to 468. Awesome. So once again, that's when we rearrange to isolate and solve for X all by itself. So if we want to do it in pieces, we will note that negative 0.20 multiplied by 60 will be simply equal to -12, and -12 will be equal to X minus 480. So that would mean that X is equal to 480 minus 12, which once again is X is equal to. 468. Awesome. So with that in mind, now that we know that our final answer is 468, looking at our multiple choice answers, that means our final answer, without a doubt has to be multiple choice answer letter A, which once again is 468. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Normal Distribution

Percentiles

Z-scores

Watch next