Question

When the manufacturing process is working properly, NeverReady batteries have lifetimes that follow a slightly right-skewed distribution with µ = 7 hours. A quality control supervisor selects a simple random sample of n batteries every hour and measures the lifetime of each. If she is convinced that the mean lifetime of all batteries produced that hour is less than 7 hours at the 5% significance level, then all those batteries are discarded.

a. State appropriate hypotheses for the quality control supervisor to test.

b. The supervisor wants to conduct a test using samples of only n=4 so as to avoid wasting batteries. Explain briefly why a sample size of 30 would be necessary.

c. Suppose you draw a random sample of 30 batteries and find that the mean battery life for the sample is 6.8 hours and the standard deviation of the sample is 0.6 hours. Conduct a hypothesis test at the 0.05 level to determine if there is sufficient evidence to conclude that the mean lifetime of all batteries is less than 7 hours.

d. Describe a Type I and a Type II error in this situation and the consequences of each.

please show work!!

Answer 1

a) Ho: µ=7 ; Ha: µ<7

b) A good sample is one which truly represents the charateristics of the population. If one choses lesser number of items from a population then there is a possibility that they may deviate from the characteristics of population. This may lead to wrong results. Similarly in the above case, if the supervisor selects only 4 batteries, there are possibilities of Type I and II errors. A sample size of 30 is necessary to make sure that the distribution follows a normal distribution.

c) Null hypothesis : The mean lifetime of all the batteries is less than 7 hours.

Alternative hypothesis : The mean lifetime of all the batteries is greater than 7 hours.

Given, n = 30

µ = 6.8 hours

σ = 0.6 hours

Now if we check the value of t at 5% significance level then we get t=1.313, which is greater than the above calculated value of t (0.33). Hence we accept the null hypothesis i.e. the mean lifetime of all the batteries is less than 7 hours.

d) Type I error : The supervisor is convinced that the mean lifetime of the batteries is less than 7 hours and discards them but the process was working really fine.

Type II error : The supervisor gets convinced that the process is fine but the mean lifetime of the batteries is less than 7 hours.

The consequence of the Type I error would be the loss of money since the good batteries are being thrown out and the consequence of the Type II error would result in the trade of bad batteries which would result in customer insatisfaction and it will ruin the name of the company.