Explain in detail

The performance of a company's staff is measured by a score based upon a variety of assessment methods and criteria. The higher the score the stronger the performance of the member of staff. (a) Analysis has shown that the mean score achieved by staff is 140 with a standard deviation of 10. Let the random variable X be the score achieved by a staff member. (i) Find the standard error of the sample mean for a sample of size n = 40. [2] (ii) Calculate the probability that the sample mean will be less than 138 for a sample of size n = 40. [4] (iii) Interpret your answer to part (a) (ii). [2] (iv) How would your answer to part (a) (ii) change if you were told that the sample size was n = 160? Justify your answer, it is not necessary to perform the calculation. [2] (b) In order to improve staff performance a random sample of 42 staff have been sent on a training programme devised by the company and the performance score for each staff member calculated. A second random sample of 42 members of staff who have not been on the training programme have also had their performance scores calculated. The results of the experiment are presented below. Untrained Scores Sample mean score = 138 Sample standard deviation = 12 Trained Scores Sample mean score 146 Sample standard deviation = 16 (i) Perform a hypothesis test with 1% level of significance to determine if the train- ing course improves staff performance on average. [12] (ii) Provide a recommendation to the company on how they could improve on the testing procedure used, if they were to evaluate the benefit of the training pro- gramme again, Justify your recommendation. [3]