Q. 4 - The Practice of Statistics for AP

Question

A $10$-question multiple-choice exam offers $5$ choices for each question. Jason just guesses the answers, so he has a probability $1/5$ of getting any one answer correct. You want to perform a simulation to determine the number of correct answers that Jason gets. One correct way to use a table of random digits to do this is the following:

(a) One digit from the random digit table simulates one answer, with 5$=$ right and all other digits $=$ wrong. Ten digits from the table simulate $10$ answers. (b) One digit from the random digit table simulates one answer, with 0 or 1 $=$ right and all other digits $=$wrong. Ten digits from the table simulate $10$ answers.

(c) One digit from the random digit table simulates one answer, with odd $=$ right and even $=$ wrong. Ten digits from the table simulate $10$ answers.

(d) Two digits from the random digit table simulate one answer, with $00$ to $20$ $=$ right and $21$to $99=$ wrong. Ten pairs of digits from the table simulate $10$ answers.

(e) Two digits from the random digit table simulate one answer, with $00$ to $05=$ right and $06$ to $99$ $=$ wrong. Ten pairs of digits from the table simulate $10$ answers. 

Step-by-Step Solution

Answer

One correct way to use the random digits is option (b) $0$ or $1=$right and all other digits$=$wrong.

Step 1: Given information

A$10$-question multiple-choice exam offers $5$ choices for each question.

He has a probability $1/5$ of getting any one answer correct.

Determine the number of correct answers he gets.

Step 2: Explanation

The probability of a correct answer needs to be $1/5$

(a) Incorrect, because $5$ of the $10$ possible digits correspond with a right answer, which corresponds with a probability of $5/10=1/2$

(b) Correct, because$2$ of the$10$ possible digits correspond with a right answer, which corresponds with a probability of $2/10=1/5$

(c) Incorrect, because $5$of the $10$ possible digits correspond with a right answer, which corresponds with a probability of$5/10=1/2$

(d) Incorrect, because repeats are being ignored and thus the probability of obtaining a correct answer is then not the same for each question (while this probability should be constant).

(e) Incorrect, because$21$ of the$100$ possible pairs of digits correspond with a right answer, which corresponds with a probability of $21/100$ instead of $1/5=20/100.$