Twenty motors were put on test under a high-temperature setting. The lifetimes in hours of the motors under these conditions are
given below. Suppose we assume that the lifetime of a randomly selected motor under these conditions, X, has a Gamma (1, theta) distribution.
(a) Derive the maximum likelihood estimate of 9 and evaluate it numerically.
(b) Obtain the sample median of the data, which is an estimate of the median lifetime of a motor. What parameter is it estimating (i.e., determine the median of X)?
(c) Based on the mle, what is another estimate of the median of X?
(d) Obtain a large sample approximate 95% confidence interval for the mean lifetime of a motor.
(a) Derive the maximum likelihood estimate of 9 and evaluate it numerically.
(b) Obtain the sample median of the data, which is an estimate of the median lifetime of a motor. What parameter is it estimating (i.e., determine the median of X)?
(c) Based on the mle, what is another estimate of the median of X?
(d) Obtain a large sample approximate 95% confidence interval for the mean lifetime of a motor.
1 answer:
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If 60 is a factor of n², then √60 is a factor of n. However, n is a whole number, so its factors are whole numbers.
Simplify √60:
If 2√15 is a factor of a whole number n, then √15 must be another factor to make it a whole number.
If 60 is a factor of n², then 2, 3, and 5 must be factors of n. The factors of n² are the squares of the factors of n, so 2, 2, 3, 3, 5, and 5 must be factors of n².
Now, if 2, 2, 3, 3, 5, and 5 are factors of n², then:
* 5×5=25 must also be its factor
* 2×2×3×3=36 must also be its factor
* 2×2×5×5=100 must also be its factor
Only 16 may not be a factor of n². The answer is A.
Simplify √60:
If 2√15 is a factor of a whole number n, then √15 must be another factor to make it a whole number.
If 60 is a factor of n², then 2, 3, and 5 must be factors of n. The factors of n² are the squares of the factors of n, so 2, 2, 3, 3, 5, and 5 must be factors of n².
Now, if 2, 2, 3, 3, 5, and 5 are factors of n², then:
* 5×5=25 must also be its factor
* 2×2×3×3=36 must also be its factor
* 2×2×5×5=100 must also be its factor
Only 16 may not be a factor of n². The answer is A.