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1) Consider a study that aims to estimate the age of junior students of a very large university in the United States

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1) Consider a study that aims to estimate the age of junior students of a very large university in the United States. For this, a sample of 2,000 junior students is selected, and the sample average is 22 years and the standard deviation is 8 years. Using this information, estimate the average age of all junior students of the university, the accuracy of this estimate within a given CI with confidence level of 95%, and the margin of error.

2) Using computer software like MATLAB, we simulate (generate) a set of normally distributed data points. How could we calculate the MLE of the parameters mean μ and standard deviation σ , for the normal distribution used, if we randomly select three points from this set, say 5, 7.3, and 12.8? Let us use the interval estimation to estimate a normal population’s parameter mean, μ, assuming that the value of the standard deviation, σ , is known.

3) We present this example to show the algorithmic method of finding the MLE. Let us consider a Bernoulli random variable X with p as the probability of success and 1 −p as the probability of failure. The probability p, as a population parameter, can be thought of as a proportion of an element in a population with a special feature, say with higher education, for example. What is the MLE for p?

4) If a population is known to follow a normal distribution but the mean and variance are unknown, the MLE can be used to estimate them using a limited sample of the population, by finding particular values of the mean and variance so that the observation is the most likely result to have occurred. For the sake of quality control, a small company selects a random sample of ten staff members from total workers to test the effectiveness of the jobs they are performing. The question is for what value of p (as the probability of success) a sample may most likely look like the one observed?

5) Suppose a manufacture producing cellphone covers produces 55% of its products for a particular location in pink, aiming at female customers in that location. To test the market, the marketing department selects a random sample of size 300 ladies from that location. We want to find the probability that more than 60% of the sample elements favor the color pink for their cellphone covers. Show that the sample mean is a consistent estimator of the population mean.

6) A very destructive Hurricane Katrina originated over the Bahamas on August 23, 2005, as Category 5 hurricane over the warm waters of the Gulf of Mexico and made its first landfall in the United States of America on Florida and Louisiana. However, it weakened to Category 5 hurricane and made a second landfall on August 29, 2005, over Southeastern Louisiana and Mississippi causing catastrophic and devastating fatal and financial damages, partially as a result of fatal engineering flaws in the flood protection system known as levees around the city of New Orleans. The total property damage was estimated as $125 billion. The damages included oil spills estimated over 10,000 gallons (38,000 L). The oil spills, in gallons, reported by nine company’s locations are as follows:991,000; 3,780,000; 13,440; 819,000; 53,000; 1,050,000; 25,000; 461,000; 13,000.We want to find the estimate of the mean and SE of all oil spills as a result of Hurricane Katrina in August 2005.