A smog alert is issued when the amount of particulate matter

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.

Solution

We have given that A smog alert is issued when the amount of particulate matter in the air is found to be greator than 7ppm.

The hypothesis for the test is,

H0 : µ = 7 Vs H1 : µ > 7

Since the test is right sided.

Here we have given n = number of stations = 16

mean (Xbar) = 7.8 ppm

standard deviation (s) = 2 ppm

Here we have given the information regarding sample and population standard deviation is unknown.

So we use t-test here.

The test statistics for testing mean is,

t = (Xbar - µ) / [ s/sqrt(n) ]

= ( 7.8 - 7) / [ 2 / sqrt(16) ]

= 0.8 / 0.5 =1.6

alpha = 0.05 (level of significance)

P-value :

We can calculate P-value by using EXCEL ;

=tdist(x, deg_freedom, tails)

where x is the test statistic value

deg_freedom = n - 1

tails = 1

P-value = 0.065

Since P-value > alpha

fail to reject H0 at 5% level of significance.

Conclusion : There is sufficient evidence to say that A smog alert is issued when the amount of particulate matter in the air is 7 ppm.

From conclusion here is risk of type II error.

because Type II error = P(accept H0 / H1 is true)

95% confidence interval for mean is ,

Xbar - E < µ < Xbar + E

where E is margin of error.

E = tc * s / sqrt(n)

tc is the critical value

EXCEL : =tinv(probability,deg_freedom)

where probability = 2*alpha

tc = 1.753

E = 1.753 * 2 / sqrt(16) = 0.8765

lower limit = Xbar - E = 7.8 - 0.8765 = 6.9235

upper limit = Xbar + E = 7.8 + 0.8765 = 8.6765