Question

In: Statistics and Probability

A student hasn’t studied at all and therefore guesses each answer uniformly at random, independently of all the other answers. Define the following random variables:

• R: the number of answers the student gets right

• W: the number of answers the student does not get right

• S: the student’s score on the test

Q1a:

What is the distribution of R? Either state the possible values and provide a formula for the probabilities, or provide the name and parameters of the appropriate distribution. Explain your answer.

Q1b:

Find E(R) and SD(R).

Q1c:

True or False: SD(R) = SD(W). Explain your answer

Q1d:

Find E(S).

Q1e:

Find SD(S).

Expert Solution

each with five possible answers of which one is right. A student hasn’t studied at all and therefore guesses each answer uniformly at random. The probability that the student gets right the answer to any given question is 1/5=0.20

Q1a: What is the distribution of R? Either state the possible values and provide a formula for the probabilities, or provide the name and parameters of the appropriate distribution. Explain your answer.

Let R be the number of answers the student gets right in a test of 100 questions. We can say answering each question is a Bernoulli trial with 2 outcomes, get it right (success) or do not get it right (failure), with a success probability of 0.20. Since guesses each answer uniformly at random, independently of all the other answers, we can say that answers to 100 questions is equivalent to doing 100 independent and identical Bernoulli trials.

ans: We can say that R has a Binomial distribution with parameters, number of trials (number of questions in te test) n=100, and success probability (The probability that the student gets right the answer to any given question) p=0.20.

That is

The probability mass function of R is

Q1b:Find E(R) and SD(R).

Using the formula for Binomial distribution, the expected value of R is

The standard deviation of R is

ans: E(R)=20, SD(R)=4

Q1c: True or False: SD(R) = SD(W). Explain your answer

We can write W=100-R

The variance of W is

Hence SD(W)=SD(R)

ans: True SD(R) = SD(W). As W= 100-R and hence SD(W)= SD(100-R)=SD(R)

Q1d: Find E(S).

We can write W as

W=100-R

We can write the score S as

The expected value of S is

ans: E(S)=0

Q1e: Find SD(S).

The variance of S is

The SD of S is

ans: SD(S)=20

orchestra answered 2 years ago

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