In a chemical process a solution is used that has a given pH value. The pH is de

Question

In a chemical process a solution is used that has a given pH value. The pH is determined by a method which is such that if used on random samples from a solution with pH equals to , the measurement results will be independent and normaly distributed ie. the measurement results are N (, 0.052).

A) How large is the likelihood of a single determination of the pH value should differ more than 0.01 from the true pH value ()?

B) Let D denote the difference between two samples, X1 and X2, D = X1 - X2. Draw the probability density of D and show the probability (as area) for D being less than -0.01 or greater than 0.01.

C) What is the probability of the difference between two such pH determinations of random samples from the same resolution, D, being greater than 0.01?

D) Four samples of the solution have been taken and have obtained the following pH values: 7.58 7.49 7.62 7.55. Use this to estimate the unknown and true PH value. Find dot estimate and determine a 95% confidence interval for .

E) The most favorable pH value of the process is 7.50. Do you, based on calculations in d) question that the true pH value can be 7.50? Explain.

F) How large must n (number of measured pH values) be at least in order for the confidence interval length to not exceed 0.05?

Explanation / Answer

(a)

For a difference of 0.01 from the mean pH value, the z-score is:

z = 0.01/0.052 = 0.192

Looking from the z-table, the p-value for this z-score is:

p = 0.423

Since the pH value can be obtained on either side of the mean, so the required likelihood is two times this probability value

So, likelihood = 2*p = 0.846 or 84.6%

(d)

For the given sample,

Sample mean, m = 7.56

Sample size, n = 4

Population standard deviation, S' = 0.052

Standard error, S'' = S'/n0.5 = 0.052/40.5 = 0.026

For 95% CI, critical z-score, zc = 1.96

So, 95% CI is:

m-zc*S'' < < m+zc*S''

Putting values and solving we get:

7.56-1.96*0.026 < < 7.56+1.96*0.026

7.509 < < 7.61

(e)

Since 7.50 is not in the specified interval, so this is not an acceptable value for true pH.

(f)

Length of 95% CI= 2*zc*S'' = 2*zc*(S'/n0.5) = 2*1.96*(0.052/n0.5) = 0.204/n0.5

Now we want this to be within 0.05

So, 0.204/n0.5 <= 0.05

Solving we get:

n => 16.64 or approximately n >= 17

So, n should atleast be 17

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