Chapter 5: Q. 9 (page 309)

Will Luke pass the quiz ? Luke’s teacher has assigned each student in his class an online quiz, which is made up of $10$ multiple-choice questions with $4$ options each. Luke hasn’t been paying attention in class and has to guess on each question. However, his teacher allows each student to take the quiz three times and will record the highest of the three scores. A passing score is $6$ or more correct out of $10$ . We want to perform a simulation to estimate the score that Luke will earn on the quiz if he guesses at random on all the questions.
a. Describe how to use a random number generator to perform one trial of the simulation. The dotplot shows Luke’s simulated quiz score in $50$ trials of the simulation.
b. Explain what the dot at $1$ represents.
c. Use the results of the simulation to estimate the probability that Luke passes the quiz.
d. Doug is in the same class and claims to understand some of the material. If he scored $8$ points on the quiz, is there convincing evidence that he understands some of the material? Explain your answer.

Short Answer

Expert verified

a. When guessing, we have a one in four probability of selecting the proper answer choice on a question because each question contains four answer alternatives, one of which is correct.

b. One trial resulted in a quiz score of one, as indicated by the dot at $1$ . Alternatively, in the simulation, there was one trial with one correct answer to each of the ten questions.

c. There are now ten multiple choice questions, each with four options. If you answer six or more questions properly, you will pass the test. In the simulation, the dots in the dot plot represent the simulated quiz score on $50$ trials.

d. The dots in the dot plot represent the simulated quiz score over $50$ trials, assuming the person is guessing the answer to each question.

Step by step solution

Part (a) Step 1 : Given information

We have to describe how to use a random number generator to perform one trial of the simulation.

Part (a) Step 2 : Simplification

Luke's teacher wants to administer a test to his students by asking multiple choice questions. To complete one simulation trial, we must now use a random number generator. First, we'll go over

10

multiple-choice questions, each with four options. It is assumed that you will pass if you accurately answer six or more questions. When guessing, we have a one in four probability of selecting the proper answer choice on a question because each question contains four answer alternatives, one of which is correct. Let's generate

10

digits from one to four using a random number generator, where one indicates a correct response and

2, 3, 4

represents a wrong answer.

Part (b) Step 1 : Given information

We have to explain what the dot at $1$ represents.

Part (b) Step 2 : Simplification

The question includes a dot plot that depicts Luke's simulated quiz score. In the simulation, the dots in the dot plot represent the simulated quiz score on

50

trials. One trial resulted in a quiz score of one, as indicated by the dot at

1

. Alternatively, in the simulation, there was one trial with one correct answer to each of the ten questions.

Part (c) Step 1 : Given information

We have to use the results of the stimulation to estimate the probability that Luke passes the quiz.

Part (c) Step 2 : Simplification

We need to figure out what the chances are that Luke will pass the quiz. There are now ten multiple choice questions, each with four options. If you answer six or more questions properly, you will pass the test. In the simulation, the dots in the dot plot represent the simulated quiz score on $50$ trials. The fact that five dots are above six and none are to the right of six means that $5$ of the $50$ trials result in a quiz score of at least $6$ . As a result,

$\frac{5}{50} = \frac{1}{10}$ $= 0.10$

As a result, Luke's chances of passing the quiz are approximately $0.10$

Part (d) Step 1 : Given information

We have to explain is there a convincing evidence that he understands some of the material.

Part (d) Step 2 : Simplification

The dots in the dot plot represent the simulated quiz score over

50

trials, assuming the person is guessing the answer to each question. We can see that there are no dots above

8

or to the right of

8

on the dot plot, implying that getting a quiz score of at least

8

when guessing at random is extremely rare. This indicates that there is strong evidence that he comprehends part of the subject, as it is doubtful that he answered every question with a guess.