Question #2823

Dr. Al Kali, an environmental chemist, measures the pH of numerous samples of water collected from a polluted lake. His data are normally distributed with a mean pH of 8.25 and a standard deviation of 0.35. What should Dr. Kali report as the margin of error for the 90% confidence interval for his data?

Expert's answer

Let X ne the pH of numerous samples of water collected from a polluted lake.

By assumption X has normal distribution N(8.25,0.35).

Then Z=((X-8.25)/0.35) has normal distribution N(0,1).

Then margin of error is equal to

δ = 0.35 * z, where z is such that

P(-z< Z < z) = 0.9

If Φ is a c.d.f. for N(0,1), then

z= - Φ^{-1} ((1-0.9)/2)= -Φ^{-1} (0.05)= 1.645

Hence

δ = 0.35 * 1.645 = 0.576

By assumption X has normal distribution N(8.25,0.35).

Then Z=((X-8.25)/0.35) has normal distribution N(0,1).

Then margin of error is equal to

δ = 0.35 * z, where z is such that

P(-z< Z < z) = 0.9

If Φ is a c.d.f. for N(0,1), then

z= - Φ

Hence

δ = 0.35 * 1.645 = 0.576

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