# A Firm Produces Batteries That Have A Lifetime Which Is Normally Distributed Wit

A firm produces batteries that have a lifetime which is normally distributed with a mean of 320 minutes and a standard deviation of 20 minutes. The firm needs to keep an eye on the production process to ensure that everything is working properly and that batteries are not being produced that do not meet the advertised standard. This is done by calculating the mean of the sample. To do this they regularly select a sample of 36 batteries in order to test the process.

(a) Describe the sampling distribution of the sample mean lifetime of batteries

(b) State a range within which you would expect the middle 85% of the sample means to lie.

(c) If the process were working correctly, what is the probability that a sample would produce a mean of less than 345 minutes?

(d) Based on your answer to part (c), what would you conclude about the process if the sample produced a mean life of batteries of 350 minutes?

(e) What is the probability that three samples in a row would have a mean life time of less than 345 minutes?

(f) If you were to change the sample size to 20, state the assumptions that would be needed and also the effects that could be seen re the results of parts (b) to (e) is these assumptions were to be violated. You may need to draw some diagrams to justify some of your statements.