Q1. A rare event occurs in a large population with probability 0.0004 per individual per year. In a city of 20,000 individuals, the event is tracked annually. (a) Using the Poisson approximation, compute the probability that in a given year, at least 10 but no more than 15 individuals experience the event. (b) If the city is divided into 4 equal districts, and the event occurrences are independent, what is the probability that at least one district records at least 5 occurrences in the same year? Show all steps, including justification for the use of the Poisson approximation and all intermediate calculations. (10 Marks)
Q2. A light bulb’s lifetime (in hours) is normally distributed with unknown mean \mu and known standard deviation =50 hours. A sample of 35 bulbs shows an average lifetime of 1200 hours. The company wants to ensure that at least 90% of bulbs last more than 1100 hours. Find the maximum mean lifetime that satisfies this and check if the sample supports this claim. (10 Marks)
Q3 (B) Suppose you are given a dataset of 10 observations where the independent variable X is the monthly advertising spend (in $1000s) and the dependent variable Y is the monthly sales (in $10,000s). The regression equation Y = a + bX is fitted, and the following is known: the sum of squared esiduals (SSE) is 180, the total sum of squares (SST) is 600, and the explained sum of squares (SSR) is 420.
(a) Calculate the coefficient of determination (R²) and interpret its meaning.
(b) If the standard error of the regression is required for a 95% confidence interval for a forecast at X = 15, compute the standard error given n = 10 and k = 1.
(c) If the regression equation is Y = 2.5 + 1.8X, estimate the 95% confidence interval for the predicted sales when X = 15, using z = 1.96.
Show all steps and justify the use of each value. (5 Marks)
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