You want to estimate the probability that a lightbulb of a famous brand will last more than 6.500 hours. You choose 10,000 bulbs randomly and observe that 9,580 did last more than 6,500 while the rest did not.

a. Write down the likelihood function to estimate this probability. Be specific with the number of positive test results in the function. (4 points)
b. What is the value of the maximum likelihood estimate of the probability? (4 points)
c. What is the likelihood of observing this experiment outcome when the true probability is 98%? (4 points)
d. What is the method of moments estimate? (4 points)
e. Argue whether the following statement is true or false: In general, the method of moments estimate is always equal to maximum likelihood estimate. (4 points)

Answer/Step-by-step explanation:

(a) The likelihood function to estimate this probability can be written as:

mat[1000, 9800]p9580(1 - p)420

(b) The value of the maximum likelihood estimate of the probability 0.958(By taking log of expression in (a) above)

(c) when the true probability is 98%, then it implies that 9800 of 10,000 bulbs did last over 6500hours.

Therefore, the likelihood is p(9800) = mat[10000, 9800]p9800(1 - p)200

(d) Method of moments estimate is the estimation of all the parameters of the population sample.

(e) The statement is FALSE because estimates by the method of moments are not necessarily sufficient statistics, because sometimes fail to take into account all relevant information in the sample. As in the above question

use distributive property on all like terms and change your signs if you have a double negative or if you have positive/negative