Each section of the SAT test is supposed to be distributed normally with a mean of 500 and a standard deviation of 100. Suppose 5 students in a class took the SAT math test. They received the following scores: 400, 450, 575, 600, and 625.
(a) Assuming a normal distribution, write out the likelihood function for estimating the mean μ and standard deviation σ of the average class score.
(b) Calculate the MLE estimate for μ and the standard deviation (σ) for this class.
(c) What is the value of the log-likelihood function at this estimate?
(d) What is the least-squares estimate for μ and σ in your data? Are there any differences? Why or why not?
(e) Using a likelihood-ratio test, test whether the MLE estimates from the class are significantly different from the national mean of 500 and national standard deviation of 100.
(Hint: Start by plugging those values into your log-likelihood function. Remember you are testing two restrictions).