Question

A student takes a 48 question multiple choice quiz. The student is unprepared, so they guess on every question. The probability of guessing a correct answer is 1/4. Find the mean and standard deviation for the random variable

Answer 1

`Mean = 12`

`Standard\ Deviation = 6`

Explanation:

Given

Number of questions, n = 48

Probability of correct answer, p = 1/4

Required

- Find the mean

- Find the standard deviation

Using probability notations;

n = 48 and p = 1/4

Mean is calculated as follows;

`Mean = np`

Substitute 48 for n and 1/4 for p

`Mean = 48 * (1)/(4)`

`Mean = (48)/(4)`

`Mean = 12`

The Mean is 12

Calculating Standard Deviation (SD)

`SD = √(np(1-p))`

Substitute 48 for n and 1/4 for p

`SD = \sqrt{48 * (1)/(4)(1-(1)/(4))}`

Solve the expression in the bracket

`SD = \sqrt{48 * (1)/(4)((4-1)/(4))}`

`SD = \sqrt{48 * (1)/(4)((3)/(4))}`

Open Bracket

`SD = \sqrt{48 * (1)/(4)*(3)/(4)}`

`SD = \sqrt{(48 * 1 * 3)/(4)}`

`SD = \sqrt{(144)/(4)}`

`SD = √(36)`

Take Square Root of 36

`SD = 6`

The standard deviation is 6