Question

A market research company employs a large number of typists to enter data into a computer database. The time taken for new typists to learn the computer system is known to have a Normal distribution with a mean of 130 minutes and a standard deviation of 20 minutes. A candidate is automatically hired if he or she learns the computer system in less than 100 minutes. A cut-off time is set at the slowest 40% of the learning distribution. Anyone slower than this cut-off time is not hired.

a) The proportion of new typists that take under two hours to learn the computer system is? (answer is 0.3085)

b)What proportion of candidates will be automatically hired? (answer is 0.0668)

c)What is the cut-off time the market research company uses? (answer is 2 hours 15 minutes)

d) You sample 30 typists, what is the sampling distribution of the mean time taken for the new typists to learn the computer system? (answer is  x¯ ∼ N(130, 3.65) )

e)What is the probability that the mean time taken for the sample of 30 typists to learn the computer system is less than 120 minutes? (answer is  0.0031)

(CAN YOU PLEASE SOLVE WITH PROPER EXPLAINATION AND MAKE SURE THE ANSWER IS THE SAME AS THE ONE GIVEN IN BRACKETS FOR ALL PARTS, THANK YOU)

Answer 1

Here we have

a)

2 hours = 2*60 minutes= 120 minutes

The z-score for X = 120 is

The proportion of new typists that take under two hours to learn the computer system is

P(X < 120) = P(z < -0.50) = 0.3085

b)

The z-score for X = 100 is

The proportion of candidates will be automatically hired is

P(X < 100) = P(z < -1.5) = 0.0668

c)

Here we need z-score that has 0.40 area to its left. using z table, z-score -0.25 has approximately 0.40 area to its left. So required X is

That is required cutoff is 125 minutes or 2 hours 5 minutes.

d)

Sample size: n=30

The sampling distribution of sample mean will be approximately normal distribution with mean

and standard deviation is

e)

The z-score for is

The probability that the mean time taken for the sample of 30 typists to learn the computer system is less than 120 minutes is