Fine particulate matter (PM2.5) is consider harmful if the average concentration is more than 12 microgram per cubic meter in a year. One research on air quality randomly sampled 36 locations in LA county and measured the PM2.5 concentration. The following information is obtained for PM2.5 concentration (in microgram per cubic meter). The average PM2.5 is 14 and sample standard deviation is 6. No outlier is observed. A previous study claimed the national average is 10.

(1) Write down the null and alternative hypotheses that would be appropriate for testing if the PM2.5 concentration in LA county is higher than the national average. (2 pts)

(2) Clearly state and check all conditions necessary for inference on these data. (4pts)

(3) Regardless of your answer in (b), test the hypothesis with 5% significance level that PM2.5 concentration in LA county is higher than the national average and draw your conclusion. (6pts)

You can find useful t values and z values that cut an upper tail of 5%.

z = 1.645, t(df=35) = 1.689, t(df=36) = 1.688, t(df=37) = 1.687

You can find useful t values and z values that cut an upper tail of 2.5%.

z = 1.96, t(df=35) = 2.030, t(df=36) = 2.028, t(df=37) = 2.026