History Grades In a history class, the grades for various asse...

Question

History Grades In a history class, the grades for various assessments are all positive numbers and have different distributions. Determine whether the grades for each assessment could be normally distributed. Explain your reasoning.

e. an extra credit assignment with a mean of 2.25 and a standard deviation of 2.49

e. an extra credit assignment with a mean of 2.25 and a standard deviation of 2.49

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says, in a chemistry class, the bonus points for a small extra credit quiz are all positive numbers and vary from student to student. Determine if the scores for this quiz can reasonably follow a normal distribution. Provide your reasoning. And we're given for this quiz, the mean bonus score is 1.8, with the standard deviation of 2.1. We're also given 4 possible choices as our answers. For choice A, no, because the standard deviation is too small compared to the mean. For choice B, yes, because the mean and standard deviation are both positive. For choice C, we have yes because the data fits exactly the bell-shaped curve for a normal distribution. And for choice D, no, because the data would include negative values which are not possible for bonus scores. So in this problem we're asked to determine if the scores for this quiz could reasonably follow a normal distribution. Now we call for a normal distribution, that that distribution is symmetric and continuous with values that exceed in both the positive and negative directions about the mean. Also for a normal distribution, the data can be expected to range roughly from mu. -3 sigma. 2 Me plus 3 sigma. Where mu here is our mean and sigma is our standard deviation. And what we're going to do is calculate the upper and lower bounds for this range, using the information that we're given in the problem. So we're gonna start off with the upper bound. And here we're gonna have you, which is our mean, and we're told that our mean, or our quiz is 1.8, so we'll have 1.8. Plus, 3, multiplied by Sigma, which is our standard deviation, and we're told that it has a value of 2.1. And so when we evaluate this expression, this is going to be equal to 8.1. We'll do something similar for the lower bound. So here we'll have mu, which is our mean of 1.8 minus 3, multiplied by our standard deviation sigma of 2.1. And when we evaluate this expression, this is going to be equal to -4.5. So here our lower bound is negative, which means that in order to have a normal distribution, We need to have negative values in our data. However, we're told that all of our bonus scores are all positive numbers. Therefore, we have a contradiction if we have a normal distribution. So, no. The scores for this quiz do not reasonably follow a normal distribution. So, based on this evaluation, if we look at our answer choices, we see that the correct answer choice actually corresponds to answer choice D, which says no, because the data would include negative values, which are not possible for bonus scores. 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

Mean and Standard Deviation

Assessment Distribution

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