A random sample of n bike helmets manufactured by a certain company is selected.

Question

A random sample of n bike helmets manufactured by a certain company is selected. Let X = the number among the that are flawed, and let p = P(flawed). Assume that only X is observed rather than the sequence of S's and Fs. (a) Derive the maximum likelihood estimator of p. (Use x for x and n for n as necessary.) If n = 16 and x = 4, what is the estimate? (Give answer accurate to 3 decimal places.) (b) Is the estimator of part (a) unbiased? yes no (c) If /7 = 16 and x-4, what is the mle of the probability (1 -p)s that none of the next five helmets examined is flawed? (Give answer accurate to 3 decimal places.)

Explanation / Answer

a)

L(p;x)=n!/(x!(nx)!) *p^x*(1p)^(nx)

which, except for the factor n!x!(nx)!, is identical to the likelihood from n independent Bernoulli trials with x=i=1nxi. But since the likelihood function is regarded as a function only of the parameter p, the factor n!x!(nx)!is a fixed constant and does not affect the MLE. Thus the MLE is again p

=x/n, the sample proportion of successes.

b) MLE = 4/16=0.25

Esitamator is unibased.

c) MLE = (1-1/4)^5 =(3/4)^5 = 0.2373

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