A smog alert is issued when the amount of particulate matter in the air is found

Question

A smog alert is issued when the amount of particulate matter in the air is found to be greater than 7 ppm. Samples collected from 16 stations have a mean of 7.8 ppm and a standard deviation of 2 ppm. a. Is there enough evidence to issue a smog alert? b. Given your conclusion, are you at risk of making a Type I error or a Type II error? Provide your answer in the context of the problem. c. Report a 95% one-sided confidence interval for the mean particulate matter in the air. d. Perform a post-hoc power analysis. Compute the difference for which you have 80% power with a sample size of n = 16 stations. e. What assumptions have you had to make? How could you go about checking this assumption?

Explanation / Answer

a)

H0=7

H17

tSTAT=X-µ/S/n

      =7.8-7/2/16

      =1.6

tCRIT=+-1.753

Since, tCRIT>tSTAT, we cannot reject the null hypothesis.

b)

(Z>=7.8-7/2/16)=1.6

=1-0.9452=0.0548 is the probability of type 2 error and 0.9452 is the power of a test.

c)

LCL=7.8-(1.753)(2)/V16=7.8-0.8765=6.9235

UCL=7.8+(1.753)(2)/V16=7.8+0.8765=8.6765

Since this interval consists the hypothesised mean, we cannot reject the null hypothesis.

d)

(Z>=7.8-7/2/16)=1.6

=1-0.9452=0.0548 is the probability of type 2 error and 0.9452 is the power of a test.

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