Question

In a study on a blood disease, the normal distribution of hemoglobin values and its arithmetic mean is 12.5 and its standard deviation is 1.0. Accordingly, this type of patients:

a) What is the probability that hemoglobin values ​​in the blood will be between 11.5 and 13.0? P (11.5 <X≤ 13.0) =?

b) What symmetrical limits are the hemoglobin values ​​of 80% of patients relative to the mean?

c) What is the hemoglobin value of the patient who has hemoglobin higher than 70% of the patients?

d) How many percent of patients have hemoglobin value less than 12.2? P (X <12.2) =?

Answer 1

The values provided in the question are as below

mean = = 12.5

standard deviation = = 1.0

a) We have to find the probability that hemoglobin values ​​in the blood will be between 11.5 and 13.0

We convert this X into Z using following formula

We find the above probability using Z table of Standard Normal Curve Areas.

The probability that hemoglobin values ​​in the blood will be between 11.5 and 13.0 is 0.8185

b) We have to find the symmetrical limits are the hemoglobin values ​​of 80% of patients relative to the mean

We use the formula

For lower limit :- We find Z value corresponding probability value = 0.1000 using Z table of Standard Normal Curve Areas. We write corresponding Z value of probability 0.1003 (close to value = 0.1000)

Z = -1.28

Lower limit = 11.22

For upper limit :- We find Z value corresponding probability value = 0.9000 using Z table of Standard Normal Curve Areas. We write corresponding Z value of probability 0.8997 (close to value = 0.9000)

Z = 1.28

Upper limit = 13.78

The (11.22, 13.78) symmetrical limits are the hemoglobin values ​​of 80% of patients relative to the mean.

c) We have to find the hemoglobin value of the patient who has hemoglobin higher than 70% of the patients

We find Z value corresponding probability value = 0.7000 using Excel function

=NORMSINV(0.7000) then press Enter

Z = 0.5244

(We write answer here using 4 decimal places)

Or

(We write answer here using 2 decimal places)

[X = 13.0244 is very close to 70% than X = 13.02]

The hemoglobin value of the patient who has hemoglobin higher than 70% of the patients is 13.0244. (13.02 write answer using 2 decimal places)

d) We have to find how many percent of patients have hemoglobin value less than 12.2.

We convert this X into Z using following formula

We find the above probability using Z table of Standard Normal Curve Areas.

We write above probability 0.3821 im percentage then it is 38.21%

The 38.21% of patients have hemoglobin value less than 12.2.

Summary :-

a) The probability that hemoglobin values ​​in the blood will be between 11.5 and 13.0 is 0.8185

b) The (11.22, 13.78) symmetrical limits are the hemoglobin values ​​of 80% of patients relative to the mean.

c) The hemoglobin value of the patient who has hemoglobin higher than 70% of the patients is 13.0244. (13.02 write answer using 2 decimal places)

d) The 38.21% of patients have hemoglobin value less than 12.2.