Chemists use pH to measure the acidalkaline nature of compou

Chemists use pH to measure the acid/alkaline nature of compounds. A large vat of mixed commercial chemicals is supposed to have a mean pH u= 6.3 with a standard deviation o = 1.9 pH. Assume a normal distribution for pH values. If a random sample of 10 readings in the vat is taken and the mean pH is computed. What is the probability that the mean pH x for these readings is:

a. u-x= and o-x=

b. P(x 5.2) = P( ) = _______________ c. P(x 7.1) = P( ) = _______________ d. P(5.2 x 7.1) = P( ) = _______________

Solution

a)
Mean ( u ) =6.3
Standard Deviation ( sd )=1.9/ Sqrt ( 10 ) = 0.6008
Number ( n ) = 10
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  
b)
P(X >= 5.2) = (5.2-6.3)/1.9/ Sqrt ( 10 )
= -1.1/0.601= -1.8308
= P ( Z >-1.8308) From Standard Normal Table
= 0.9664  
c)
P(X <= 7.1) = (7.1-6.3)/1.9/ Sqrt ( 10 )
= 0.8/0.6008= 1.3315
= P ( Z <1.3315) From Standard NOrmal Table
= 0.9085                  
d)              
To find P(a <= Z <=b) = F(b) - F(a)
P(X < 5.2) = (5.2-6.3)/1.9/ Sqrt ( 10 )
= -1.1/0.6008
= -1.8308
= P ( Z <-1.8308) From Standard Normal Table
= 0.03357
P(X < 7.1) = (7.1-6.3)/1.9/ Sqrt ( 10 )
= 0.8/0.6008 = 1.3315
= P ( Z <1.3315) From Standard Normal Table
= 0.90849
P(5.2 < X < 7.1) = 0.90849-0.03357 = 0.8749