Question

You are to take a multiple-choice exam consisting of 100 questions with 5 possible responses to each question. Suppose that you have not studied and so must guess (select one of the five answers in a completely random fashion) on each question. Let \(x\) represent the number of correct responses on the test. a. What kind of probability distribution does \(x\) have? b. What is your expected score on the exam? (Hint: Your expected score is the mean value of the \(x\) distribution.) c. Compute the variance and standard deviation of \(x\). d. Based on your answers to Parts (b) and (c), is it likely that you would score over 50 on this exam? Explain the reasoning behind your answer.

Short answer

a. \(x\) follows a binomial distribution. b. The expected score on the exam is 20. c. The variance and standard deviation of \(x\) are 16 and 4, respectively. d. It is highly unlikely to score over 50 on the exam by randomly guessing as 50 is more than seven standard deviations away from the expected score of 20.

Step-by-step solution

Type of Probability Distribution

Given that there are only two possible outcomes (correct or incorrect) for each question, and each question is independent of the others, the number of correct answers follows a binomial distribution. So, \(x\) has Binomial Distribution.

Expected Score

The expected value (mean) of a binomial distribution is given by \(np\), where \(n\) is the number of trials, and \(p\) is the probability of success. Here, \(n=100\) (number of questions), \(p=1/5\) (probability for each question, since there are 5 choices and one is correct). So, Expected score \(E(x) = np = 100 * 1/5 = 20\).

Variance and Standard Deviation

The variance of a binomial distribution is given by \(np(1-p)\). So, Variance \(Var(x) = np(1-p) = 100 * 1/5 * 4/5 = 16\). The standard deviation is the square root of the variance. So, Standard Deviation \(SD(x) = \sqrt{16} = 4\).

Probability of Scoring over 50

Given that our expected score (mean) is 20 and the standard deviation is 4, a score of 50 is significantly higher than the mean. Using the rule of standard deviations, any value over two standard deviations from the mean is considered unlikely in a normal distribution. 50 is more than seven standard deviations away from the mean (because \(50-20 = 30\), and \(30/4 = 7.5\) ). Therefore, it is highly unlikely to score over 50 on this exam by guessing randomly.

Key concepts

Binomial Distribution

When dealing with multiple-choice questions like in your exam, it's crucial to understand what kind of probability distribution you're working with. The binomial distribution perfectly applies to your scenario because it models the number of successes in a fixed number of trials, with each trial having two possible outcomes — like a correct or incorrect answer.

In the context of guessing on a multiple-choice test, each question can be seen as a 'trial.' Since there's only one correct answer out of five options, and assuming that you guess randomly, the probability of choosing the correct response (success) is always the same (\(p = \frac{1}{5}\)). Also, each question is independent of the others, which means your chance of guessing one question correctly doesn't influence your chances on another question.

Formally, a random variable like the number of correct answers on your test, denoted as x, follows a binomial distribution if there's a fixed number of trials, n, each trial is independent, and the probability of success, p, is the same in each trial. Your situation meets all these conditions with n = 100 and p = \frac{1}{5}. Therefore, the distribution of correctly guessed answers on your exam is a classic example of a binomial distribution.

Expected Value

Grasping the idea of the expected value is pivotal for understanding the performance you might anticipate on a test. It's the average outcome if an experiment (like guessing answers) was repeated a very large number of times. For any binomial distribution, the expected value, also known as the mean, is calculated by multiplying the number of trials (n) by the probability of success (p).

For your exam example, with n = 100 questions and p = \frac{1}{5}, the expected score is simply \(np = 100 \times \frac{1}{5} = 20\). This doesn't mean you'll score exactly 20; rather, if you took a vast number of such tests, on average, you'd get 20 questions right on each. The expected value serves as a central measure or a prediction of the 'center' of the distribution of probable outcomes.

Variance and Standard Deviation

Now let's dig deeper into variance and standard deviation, which many students find to be perplexing concepts. These metrics offer insight into the spread or dispersion of the possible outcomes around the expected value.

Variance provides a measure of how much the correct answers (successes) are likely to deviate from the expected number of correct answers. In a binomial distribution, you find the variance by multiplying the number of trials (n) by the probability of success (p) and the probability of failure (\(1-p\)). For the exam, variance is calculted as \(Var(x) = np(1-p) = 100 \times \frac{1}{5} \times \frac{4}{5} = 16\).

Standard deviation is simply the square root of the variance. It's a more intuitive measure of spread because it's in the same units as the original data. Applying it to our example, the standard deviation of the test scores would be \(SD(x) = \sqrt{16} = 4\). With an expected score of 20 and a standard deviation of 4, we can infer that most of your test scores would fall within a range close to 20, making it quite improbable to score over 50, which is many standard deviations away from the expected value.