Question

Suppose that

X₁, X₂,   , X_n

and

Y₁, Y₂,   , Y_n

are independent random samples from populations with means

μ₁

and

μ₂

and variances

σ₁²

and

σ₂²,

respectively. It can be shown that the random variable

U_n =

(X − Y) − (μ₁ − μ₂)

σ12 + σ22 n

satisfies the conditions of the central limit theorem and thus that the distribution function of

U_n

converges to a standard normal distribution function as

n → ∞.

An experiment is designed to test whether operator A or operator B gets the job of operating a new machine. Each operator is timed on 45 independent trials involving the performance of a certain task using the machine. If the sample means for the 45 trials differ by more than 1 second, the operator with the smaller mean time gets the job. Otherwise, the experiment is considered to end in a tie. If the standard deviations of times for both operators are assumed to be 2 seconds, what is the probability that operator A will get the job even though both operators have equal ability? (Round your answer to four decimal places.)

You may need to use the appropriate appendix table or technology to answer this question.

Answer 1

As both operators have equal ability so

As If the sample means for the 45 trials differ by more than 1 second, the operator with the smaller mean time gets the job.

The required probability is same as

As the standard deviations of times for both operators are assumed to be 2 seconds, so

Hence using normal table the required probability is