3. Each year A-Level maths students at Loreto Grammar School (LGS) are participating in the Senior Maths Challenge - with only the 1000 students with the highest scores nationwide qualifying for the next round of the competition. Trafford City Council has collected data from the last 50 years about the number of Loreto A-Level maths students qualifying for the next round. Trafford City Council is using these data to define probabilities for the number of A-Level maths students qualifying per year and secondary school in Trafford. The resulting empirical probability distribution is summarized in Table II below: Table II. Number of Loreto A-Level maths students qualifying for the next round per year Proportion of years 0 0.800 1 0.140 2 0.040 3 0.020 4 0.000 a) Verify that the distribution given in Table II presents a valid probability distribution, and calculate its mean and standard deviation. (20% of the marks for this question) b) Using the above setting, explain the relationship between sample and population parameters. How could Trafford City Council improve the accuracy of the probability estimates in Table II and why? (20% of the marks for this question) C) Assume that, at a given Trafford secondary school, qualification for the next round of the Senior Maths Challenge occurs at an average rate of 0.20 students per year. Which theoretical probability distribution may be useful in modelling the number of qualifications per year? Parameterize the distribution and calculate its mean and standard deviation. Evaluate the distribution for values of 0 to 4. Based on these results, decide whether this theoretical model provides a good approximation to the empirical probability distribution in Table II and why? (35% of the marks for this question) d) Building on the theoretical model used in (c), define a suitable probability distribution over the number of years between two qualifications. Calculate the mean and standard deviation of the distribution, and determine the probability that a given secondary school in Trafford has no next round qualification for a period of 10 years or longer. (25% of the marks for this question)