Answer the following question. Please show all your working.

Scientists have determined that benzene, a chemical solvent used in the production of plastics, can possibly cause cancer. The federal government legislated that the amount of benzene cannot be above 2.5 parts per mil- lion (ppm) in order to protect workers. You work for a steel company in Pittsburgh, which is currently being monitored to see if it meets the requirements. The government has measured the air 505 times over the past 6 months. In these observations, the average level is 2.8 ppm, and the variance is 1.8 ppm.

a. Construct a 90% confidence interval for the mean benzene levels.

b. Would a 95% confidence interval be wider or narrower than the one in your answer to (a)? Answer this question using the definition of confidence intervals and by calculating another confidence interval.

c. Test the null hypothesis that your company is meeting the federal standards, against the alternative that it is not meeting the standards. Use a level of significance ?α? = 0.05.

d. Now assume that the average level observed was instead 2.6. The sample variance is still 1.8. Assuming that the true population mean is 2.5 ppm, what is the probability that we would observe an average level at least this high? You can give a range for this probability.