When red blood cells are counted using a certain electronic counter, the standard deviation (SD)
of repeated counts of the same blood specimen is about .8% of the true value, and the distribution
of repeated counts is approximately normal. For example, this means that if the true value is
5,000,000 cells/mm3, then the SD is 40,000.
(a) If the true value of the red blood count for a certain specimen is 5,000,000 cells/mm3, what
is the probability that the counter would give a reading between 4,900,000 and 5,100,000?
(b) of the true value of the red blood count for a certain specimen is ì, what is the probability
that the counter would give a reading between .98ì and 1.02ì?
(c) A hospital lab performs counts of many specimens every day. For what percentage of these
specimens does the reported blood count differ from the correct value by 2% or more?
hallo Im Asmaa I need help with part C.
part A & B I get it
this Q is from the book and the final answer of C is 1.24% as in the book but I need to find how I reach to this answer
thanks
To answer these questions, we will use the concept of the normal distribution and the Z-score.
The Z-score measures the number of standard deviations a particular value is from the mean. In this case, we will assume that the mean is equal to the true value of the red blood count, and the standard deviation is 0.8% of the true value.
(a) To find the probability that the counter would give a reading between 4,900,000 and 5,100,000 when the true value is 5,000,000, we need to calculate the Z-scores for both values.
Z1 = (4,900,000 - 5,000,000) / (0.008 * 5,000,000)
Z2 = (5,100,000 - 5,000,000) / (0.008 * 5,000,000)
Using a Z-table or a statistical software, we can find the probabilities associated with these Z-scores. The probability is then calculated as the difference between the two probabilities:
P = P(Z2) - P(Z1)
(b) Similarly, to find the probability that the counter would give a reading between 0.98ì and 1.02ì when the true value is ì, we need to calculate the Z-scores for both values.
Z1 = (0.98ì - ì) / (0.008 * ì)
Z2 = (1.02ì - ì) / (0.008 * ì)
Again, using a Z-table or a statistical software, we can find the probabilities associated with these Z-scores. The probability is then calculated as the difference between the two probabilities:
P = P(Z2) - P(Z1)
(c) To find the percentage of specimens for which the reported blood count differs from the correct value by 2% or more, we need to calculate the Z-score for the lower value of 2% and find the probability associated with it. This represents the probability of getting a reading below this value. Then, we can subtract this probability from 1 to find the percentage for which the reported blood count differs by 2% or more.
Z = -2 / 0.008
P = 1 - P(Z)
By calculating these probabilities using Z-scores and a Z-table or statistical software, we can find the answers to these questions.