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QuestionFind the percentage of blood samples with coagulation time > 16.5 minutes, given mean = 10 min, SD = 6.5 min.

Studdy Solution

STEP 1

Assumptions1. The average coagulation time of blood is10 minutes. . The standard deviation of the coagulation time is6.5 minutes.
3. We are asked to find the percentage of blood samples whose coagulation time lasts for more than16.5 minutes.
4. We assume that the coagulation times are normally distributed.

STEP 2

First, we need to standardize the given coagulation time (16.5 minutes) to a z-score. The z-score is a measure of how many standard deviations an element is from the mean. We can calculate it using the formulaz=X−μσz = \frac{X - \mu}{\sigma}where- XX is the value we are standardizing (16.5 minutes in this case), - μ\mu is the mean (10 minutes in this case), - σ\sigma is the standard deviation (6.5 minutes in this case).

STEP 3

Now, plug in the given values for XX, μ\mu, and σ\sigma to calculate the z-score.
z=16.5−106.5z = \frac{16.5 -10}{6.5}

STEP 4

Calculate the z-score.
z=16.−106.=1z = \frac{16. -10}{6.} =1

STEP 5

Now that we have the z-score, we can find the percentage of blood samples whose coagulation time lasts for more than16.5 minutes. We can use a standard normal distribution table or a calculator with a normal distribution function to find the area to the right of z=1. This area represents the percentage of blood samples with coagulation times greater than16.5 minutes.The area to the right of z=1 is given by 1−(Z≤1)1 -(Z \leq1), where (Z≤1)(Z \leq1) is the cumulative probability up to z=1.

STEP 6

From the standard normal distribution table or using a calculator, we find that (Z≤1)=0.8413(Z \leq1) =0.8413.

STEP 7

Now, calculate the area to the right of z=1.
1−(Z≤1)=1−0.84131 -(Z \leq1) =1 -0.8413

STEP 8

Calculate the percentage of blood samples whose coagulation time lasts for more than16.5 minutes.
1−(Z≤1)=1−0.8413=0.15871 -(Z \leq1) =1 -0.8413 =0.1587This value is in decimal form, so we multiply by100 to convert it to a percentage.
Percentage=0.1587×100Percentage =0.1587 \times100

STEP 9

Calculate the final percentage.
Percentage=.1587×100=15.87%Percentage =.1587 \times100 =15.87\%So, approximately15.87% of blood samples have a coagulation time of more than16.5 minutes.

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