Question

Suppose that the pH of soil samples taken from a certain geographic region is normally distributed with a mean pH of 7 and a standard deviation of 0.10. If the pH of a randomly selected soil sample from this region is determined and equal to x, answer the following questions about it.a. What is the probability that the resulting pH is between 5.90 and 6.15?b. What is the probability that the resulting pH exceeds 6.10?c. What is the probability that the resulting pH is at most 5.95?d. What value will be exceeded by only 5% of all such pH values?

Answer 1

Answer:

a) P=0

b) P=1

c) P=0

d) X=7.1645

Step-by-step explanation:

We have the pH of soil for some region being normally distributed, with mean = 7 and standard deviation of 0.10.

a) What is the probability that the resulting pH is between 5.90 and 6.15?

We calculate the z-score for this 2 values, and then compute the probability.

$z_1=(X_1-\mu)/\sigma=(5.9-7.0)/0.1=-11\\\\z_2=(X_2-\mu)/\sigma=(6.15-7)/0.1=-8.5$

Then the probability is

$P(5.90<x<6.15)=P(x<6.15)-P(5.90)\\\\P(5.90<x<6.15)=P(z<-8.5)-P(z<-11)\\\\P(5.90<x<6.15)=0-0=0$

b) What is the probability that the resulting pH exceeds 6.10?

We repeat the previous procedure.

$z=(6.10-7)/0.1=-9\\\\P(x>6.1)=P(z>-9)=1$

c) What is the probability that the resulting pH is at most 5.95?

$z=(5.95-7)/0.1=-10.5\\\\P(x<5.95)=P(z<-10.5)=0$

d. What value will be exceeded by only 5% of all such pH values?

We have to calculate the value of pH for which only 5% is expected to be higher. This can be represented as P(X>x)=0.05.

In the standard normal distribution, it happens for a z=1.645.

$P(z>1.645)=0.05$

Then, we can calculate the pH as:

$x=\mu+z\sigma=7.0+1.645*0.1=7.0+0.1645=7.1645$