Download STA 3032, Fall 2002

STA 3032
Fall 2014
Quiz 9, Form A Solutions
1. A battery manufacturer claims that the lifetime of a certain type of battery has a population mean of 40
hours with a standard deviation of 5 hours. A random sample of 100 of these batteries is selected, and the
lifetime of each battery in the sample is measured.
a) (9 pts.) Describe, as completely as possible, the distribution of the sample mean, and give a reason for
your answer. Your answer should be in the form of a complete sentence.
Since the sample size is large (n ≥ 30), the Central Limit Theorem tells us that the distribution of 𝑋̅ is
5 ℎ𝑟𝑠
approximately normal, with a mean of 𝜇𝑋̅ = 40 ℎ𝑜𝑢𝑟𝑠 and a standard deviation of 𝜎𝑋̅ =
= 0.5 ℎ𝑟.
√10
b) (9 pts.) Find the probability that the mean lifetime for the sample is no more than 38.7 hours.
𝑃(𝑋̅ ≤ 38.7 ℎ𝑟𝑠. ) = 𝑃(−∞ < 𝑋̅ ≤ 38.7 ℎ𝑟𝑠. ) ≅ 𝑃(40 − 5 ℎ𝑟𝑠 < 𝑋̅ ≤ 38.7 ℎ𝑟𝑠. )
= 𝑛𝑜𝑟𝑚𝑎𝑙𝑐𝑑𝑓(35,38.7,40,0.5) = 0.0047.
2. (9 pts.) A random sample of size n = 2 is selected from a population having a mean of µ = 5 and a
𝑋 +2𝑋
standard deviation of σ = 1.5. Let 𝜃̂ = 1 3 2. Find the bias and variance of 𝜃̂.
The expectation of the estimator is
1
2
1
2
1
2
̂ ] = 𝐸 [ 𝑋1 + 𝑋2 ] = 𝐸[𝑋1 ] + 𝐸[𝑋2 ] = ( ) (5) + ( ) (5) = 5.
𝐸[Θ
3
3
3
3
3
3
Hence the bias is B = 0.
1
4
1
4
̂ ) = 𝑉𝑎𝑟(𝑋1 ) + 𝑉𝑎𝑟(𝑋2 ) = ( ) (1.5) + ( ) (1.5) = 0.8333.
𝑉𝑎𝑟(Θ
9
9
9
9