PLEASE ANSWER ALL PARTS ASAP!!

Suppose that 30% of people are tone deaf, and 70% of people are not.

If we present a tone-deaf person with a musical stimulus, we expect them to correctly label the stimulus 50% of the time. If we present a non-tone-deaf person with a musical stimulus, we expect them to correctly label the stimulus 75% of the time.

We select a person at random and must decide whether they are tone deaf or not. The person correctly labels 26 musical stimuli out of 40 over the course of an experiment.

1. What is the marginal likelihood of the outcome that we observed? Provide your answer as a proportion rounded to three decimal places.

Hint: Begin by identifying the data, the competing hypotheses, the priors, and the likelihood of the data under each hypothesis (you will need to use the binomial distribution in JASP to calculate the likelihood of the data under each hypothesis). Then, use the law of total probability to calculate the marginal likelihood.

2. To help us check your work, please briefly describe the numbers you used to perform the above calculation.

3. Let's use the following notation:

: the person is tone deaf

: the person is not tone-deaf

: the person correctly labels 26 musical stimuli out of 40

What is (|)? Provide your answer as a proportion rounded to three decimal places.

4. Reusing the above notation, what is (|)? Provide your answer as a proportion rounded to three decimal places.

5. Do you believe the person that we have selected is tone deaf? Briefly explain your answer using the quantities you calculated above.

6. Suppose we have three competing hypotheses, ₁, ₂ and ₃. After collecting some data , we find that

(|₁)=0.43

(|₂)=0.28

(|₃)=0.30

()=0.30

Which of the following statements is false? Hint: Begin by calculating the updating factor for each hypothesis.

a) The data provide evidence in support of ₁.

b) The data do not provide any information that changes the probability of ₃.

c) (₂|)< (₂)

d) (₁|) must be greater than both (₂|) and (₃|).

e) The data provide evidence against ₂.

f) (₁|)> (₁)