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Question 1) An employee at a café has been trained to set the coffee machine so that an espresso coffee portion results in 2

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Question 1) An employee at a café has been trained to set the coffee machine so that an espresso coffee portion results in 2.0 grams of coffee being placed into a cup. Given that the variations are expected, the employee pours eight portions. The measurements of the amounts of coffee are recorded in dataset_Assignment.xlsx.

(a) Calculate a 90% confidence interval for the mean size of espresso coffee portions. Explain the answer.

(b) Comment on whether the mean portion of coffee is equal to 2.0 grams.

Question 2) A researcher has collected the following data on a group of students, regarding whether they passed or failed an exam and whether or not they attended tutorials:

Student ID beginning with 1:

Number of students	Exam passed	Exam failed
Attended tutorials	132	27
Did not attend tutorials	120	51

 

Student ID beginning with 2:

Number of students	Exam passed	Exam failed
Attended tutorials	125	28
Did not attend tutorials	127	50

 

The researcher wants to establish whether the tutorial attendance is independent of exam success, using a chi-square test.

(a) State the hypothesis of this test.

(b) Calculate the expected frequencies for the data with the assumptions in part (a). (Present your answer in the table form).

(c) Perform the chi-square test. State your conclusion at 1% level of significance.

Question 3) A machine in a sweet factory fills bags of sweets to weigh 500 grams. The actual weight of the sweet bags is known to follow a normal distribution. The sweet manufacturer believes that the machine is under-filling the sweet bags. A sample of 10 sweet bags is taken and weighed. The dataset can be found in dataset_Assignment.xlsx.

(a) Perform a suitable t-test to determine whether the sweet bags are being consistently under-filled. State the hypotheses and the level of significance used in the test.

(b) Interpret the conclusion in part (a).

Question 4) A waiting time random variable X follows an exponential distribution with a rate l > 0 parameterized as probability density function, f(x) = le^-lx for x > 0.

The rate l has a gamma prior distribution with parameters a and b. A Bayesian credibility model provides that the posterior mean of l^-1 can be expressed as:

Zx + (1 – Z)(β/a – 1)

Where Z = m/a+m-1 and m being the number of past waiting times observed.

Assume that the parameters of the prior gamma distribution are a = 5 and β = 1.

(a) Determine an estimate of the posterior mean of l^-1 assuming m =10 by implementing 3,000 Monte Carlo repetitions.

(b) Determine an estimate of the posterior mean of l^-1 and the value of x when m = 1,000, by implementing 3,000 Monte Carlo repetitions.

(c) Plot the histograms of the samples of the posterior mean of l^-1 and of x obtained in part (b).

(d) Compare, by visual inspection of the graphs in part (c), the distributions of the posterior mean of l^-1 and the distribution of x when m = 1,000.

(e) Comment on the behavior of the credibility model as m increases with the findings in part (d).