Notation Common tests such as the SAT, ACT, LSAT, and MCAT tes...

Question

Notation Common tests such as the SAT, ACT, LSAT, and MCAT tests use multiple choice test questions, each with possible answers of a, b, c, d, e, and each question has only one correct answer. For people who make random guesses for answers to a block of 100 questions, identify the values of p, q, μ, and σ. What do μ and σ measure?

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says a student is taking a multiple choice quiz with 50 questions, each having 6 options. If the student guesses on all questions, calculate the values of PQ, mu, and sigma. Explain the significance of m and sigma. We're given four possible choices as answers. For choice A, we have P is equal to 1 divided by 6, Q is equal to 5 divided by 6, mu is equal to 8.33, and sigma is equal to 2.64. Where mu is the average number of correctly guessed answers, and sigma measures the variability or dispersion from the mean. Your choice B, we have P is equal to 1 divided by 6, Q is equal to divided by 6, m is equal to 8.33, and sigma is equal to 2.83. Great mu is the probability of guessing one question correctly, and sigma is the average number of incorrect answers. For C, we have P is equal to 1 divided by 5, Q is equal to 4 divided by 5, mu is equal to 8.33, sigma is equal to 2.74, where mu is the total number of questions, and sigma is the variance of the number of correctly guessed answers. And for choice D, we have P is equal to 4 divided by 5, Q is equal to 1 divided by 5, U is equal to 12.5, and sigma is equal to 3.79. Where mu is the highest possible score, and sigma is the error margin in guessing. Now, the first thing we need to do is calculate the values of P, Q, U, and sigma. Now, we're going to let P be the probability of guessing the correct answer for a single question. So that means P is going to be equal to the total number of possible correct options, which in this case, each question only has one correct answer, so that'll be 1, and it's going to be divided by the total number of options that we have, which is 6. So P is going to be equal to 1 divided by 6. Now Q is going to be the probability of guessing the answer incorrectly. And there is a relationship between E and Q, and here Q is going to be equal to 1 minus P, since P + Q must be equal to 1. And so that means that Q is going to be equal to 1 minus 1 divided by 6, which is going to be equal to 5 divided by 6. So these are the two probabilities that we that we were looking for, for P and Q. Next, we need to calculate mu, and mu here is our mean, and that's gonna be equal to. The total number of questions, which we will label as N, multiplied by P, which is the probability of getting an answer correct. So, this is going to be equal to. 50 Multiplied by 1 divided by 6. Which is going to be equal to. 8.33. So here we've just rounded into two decimal places. And sigma here is going to be our standard deviation. And recall that the standard deviation is going to be equal to the square root. Of the quantity Of N multiplied by P multiplied by Q. And so this is going to be equal to the square root of the quantity of 50. Multiplied by 1 divided by 6, multiplied by 5 divided by 6. And when we evaluate this expression, this is going to be equal to. 2.64, again, rounding the two decimal places. So those are the values for me and sigma. So now we have to explain the significance of mute and sigma. So, mute here is the mean value of correct answers. So that means that mu is the average number of correctly guessed answers. And sigma being the standard deviation. Means that sigma measures the variability or dispersion from the mean. So that's basically our definition of a standard deviation. And so, along with our values here, and those two explanations, we can see that the correct option for our answer choices, is going to be answer choice A. I hope that this explanation has been useful, and I'll see you in the next video.

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Key Concepts

Probability (p and q)

Mean (μ)

Standard Deviation (σ)

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