Question

The pH measurements of water specimens from various locations along a given river basin are Normally distributed with mean 8 and standard deviation 0.3. A). What is, approximately, the probability that the pH measurement of a randomly selected water specimen is greater than 8.2? B). What is, approximately, the probability that the pH measurement of a randomly selected water specimen is a value between 7.5 and 8.2? C). Find the first quartile of the distribution of the pH measurements. i.e., Find the pH value such that three-quarters of the pH measurements in this river basin are greater than this value. Please dont show how to do in excel im trying to know how to by hand, and please say where numbers come from, thank you!

Answer 1

Answer:

(a)

The probability that the pH measurement of a randomly selected water specimen is greater than 8.2 is obtained as shown below:

From the given information, the PH measurements of water specimens are normally distributed, standard deviation and mean of PH measurement are 0.3 and 8.

That is, [mu = 8], [sigma = 0.3] and [x = 8.2].

Instructions to obtain probability that the pH measurement of a randomly selected water specimen is greater than 8.2:

1.In EXCEL, Select Add-Ins > PHStat > Probability & Prob. Distributions.

2.Choose Normal.

3.In Data enter Mean as 8 and Standard deviation as 0.3.

4.In Input Options, select Probability for: X >and enter 8.2.

5.Click Ok.

Follow the above instructions to obtain the following output.

Normal Probabilities Common Data Mean Standard Deviation 8 0.3 Probability for X> X Value z Value P(X>8.2) 8.2 0.6666667 0.2525

From the output, the probability that the pH measurement of a randomly selected water specimen is greater than 8.2 is 0.2525.

That is, [Pleft( {x > 8.2} ight) = 0.2525].

(b)

The probability that the pH measurement of a randomly selected water specimen is a value between 7.5 and 8.2is obtained as shown below:

That is, [mu = 8], [sigma = 0.3].

Instructions to obtain probability that the PH measurement of a randomly selected water specimen is a value between 7.5 and 8.2:

1.In EXCEL, Select Add-Ins > PHStat > Probability & Prob. Distributions.

2.Choose Normal.

3.In Data enter Mean as 8 and Standard deviation as 0.3.

4.In Input Options, select Probability for range: and enter 7.5 <= X <=8.2.

5.Click Ok.

Follow the above instructions to obtain the following output.

Normal Probabilities Common Data Mean 8 Standard Deviation Probability for a Range From X Value To X Value Z Value for 7.5 Z Value for 8.2 P(X<-7.5) P(X<-8.2) P(7.5< X<-8.2) 7.5 8.2 1.666667 0.6666667 0.0478 0.7475 0.6997

From the output, the probability that the pH measurement of a randomly selected water specimen is a value between 7.5 and 8.2 is 0.6997.

That is, [Pleft( {7.5 < x > 8.2} ight) = 0.6997].

(c)

The PH value such that three-quarters of the pH measurements in this river basin are greater than this value is found as shown below:

The 25% of the data is below the first Quartile and 75% of the data is above first Quartile.

That is, [mu = 8], [sigma = 0.3] and .[P - { m{value}} = 0.25] .

Instructions to obtain PH value such that three-quarters of the pH measurements in this river basin are greater than this value:

1.In EXCEL, Select Add-Ins > PHStat > Probability & Prob. Distributions.

2.Choose Normal.

3.In Data enter Mean as 8 and Standard deviation as 0.3.

4.In Input Options, select X for Cumulative Percentage and enter 25.

5.Click Ok.

Follow the above instructions to obtain the following output.

Normal Probabilities Common Data Mean 8 Standard Deviation 0.3 Find X and Z Given Cum. Pctage. Cumulative Percentage Z Value X Value 0.25% 2.8070 7.1579

From the output, the PH value such that three-quarters of the pH measurements in this river basin are greater than this value is 7.1579.