A customer service helpline recruiters call operators on a probationary basis before hiring a select group as permanent employees. The average duration of the calls received by the operators is known to follow a normal distribution with a mean of 18 minutes and a standard deviation of 5 minutes. Probationary operators who maintain an average call duration between 15 minutes and 30 minutes during the probationary period get hired in permanent positions, while the rest are fired. a) What is the probability that a randomly selected operator spends more than 25 minutes on an average call? [2 marks] b) Determine the 85th percentile value for the call duration. [2 marks] c) What is the probability that a randomly selected operator will get fired after the probationary period? [2 marks] 2.A company produces high quality and long lasting batteries for cars. The CEO of the company claims that the average lifespan of a battery is 270 days with a standard deviation of 50 days under normal condition. A quality control officer randomly selected 45 batteries from the production line and tested the longevity period of these batteries. It is found that the tested batteries had a mean life span of 260 days.
a) State the appropriate hypotheses to support the claim of the CEO that the average lifetime of a battery is 270 days. [2 marks]
b) Describe Type I and Type II errors in this context, and comment on the severity of each type of error. [3 marks]
c) Test the hypotheses stated in part (a) at a 1% level of significance, and comment on your findings. [5 marks]

3.A manufacturer of running shoes knows that the average lifetime for a particular model of shoes is 15 months. A designer came with a new model of shoes which were worn by 15 individuals, and the lifetime of each of those pairs of shoes are given below:
15, 18, 17, 20, 16, 16.5, 15.5, 17.5, 19, 14, 21, 18.5, 16.5, 15, 14
a) State the appropriate hypotheses to test whether the lifetime of shoes are showing evidence of a longer lasting product than expected. [2 marks]
b) Determine the mean and standard deviation for the lifetimes of the sample of shoes. [3 marks]
c) Test the hypotheses stated in part (a) at a 5% level of significance, and comment on your findings. [5 marks

4.SwiftTech is a computer dealership that sells laptops with both Intel and AMD chips. On any given day 40% of all the customers ordering a new laptop want one with an AMD chip, while the rest prefer Intel. Suppose 18 customers were randomly selected among those shopping for laptops on a specific day.
a) What is the mean and the standard deviation of the number of customers who want an AMD laptop? [2 marks]
b) What is the probability that the number of people who want AMD laptops is more than two standard deviations away from the mean value? [4 marks]
c) SwiftTech has 10 AMD laptops and 12 Intel laptops in stock. If the 18 people come in one after another to purchase a laptop, what is the probability that all 18 will get the type of laptop they want? [4 marks]