Notation Common tests such as the SAT, ACT, LSAT, and MCAT tests use multiple choice test questions, each with possible answers of$\mathbf{a}\mathbf{,}\mathbf{b}\mathbf{,}\mathbf{c}\mathbf{,}\mathbf{d}\mathbf{,}\mathbf{e}$ , 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, $\mu$, and $\sigma$. What do$\mu$and $\sigma$measure?

A multiple-choice-based test is considered. A sample of 100 questions is answered. Each question has 5 possible answers and only one correct answer.

Let X (number of successes) denote the number of correct answers.

The probability of success (p) is computed below:

$$\begin{gathered} p=\frac{1}{5} \\ =0.2 \end{gathered}$$

The probability of failure (q) is equal to:

$$\begin{gathered} q=1-p \\ =1-0.2 \\ =0.8 \end{gathered}$$

The number of independent trials (n) is equal to 100.

By applying the normal approximation to the given setup, the following values are obtained:

$$\begin{gathered} p=0.2 \\ q=0.8 \end{gathered}$$

Here, $\mu$and $\sigma$represents the mean and population standard deviation, respectively.

$$\begin{gathered} \mu =np \\ =100\left(0.2\right) \\ =20 \end{gathered}$$

$$\begin{gathered} \sigma =\sqrt{npq} \\ =\sqrt{100\left(0.2\right)\left(0.8\right)} \\ =4 \end{gathered}$$

Therefore,

• p= 0.2
• q= 0.8
• $\mu$=20
• $\sigma$=4

Here, $\mu$ denotes the mean number of correct answers when making random guesses for 100 questions. Thus, the mean number of correct answers when making random guesses for 100 questions is equal to 20.

$\sigma$measures the standard deviation of the number of correct numbers in 100 questions. It is the amount of variation in the number of correct answers when random guesses are made for 100 questions.

Thus, the number of correct answers varies by 4.