For a group of 2,000 patients… Word problem involving standard deviation, probability, normal distribution.

"For a group of 2000 patients, the mean clotting time for blood was 8 seconds with a standard deviation of 3 seconds. The times were evenly distributed."

What percentage of the patients had a blood-clotting time between 8 and 14 seconds?

How many of the patients have a blood clotting time between 5 and 11 seconds?

What is the probability that a patient from this group had a clotting time more than 11 seconds?

For any normal random variable X with mean μ and standard deviation σ , X ~ Normal( μ , σ ), (note that in most textbooks and literature the notation is with the variance, i.e., X ~ Normal( μ , σ² ). Most software denotes the normal with just the standard deviation.)

You can translate into standard normal units by:
Z = ( X – μ ) / σ

Moving from the standard normal back to the original distribuiton using:
X = μ + Z * σ

Where Z ~ Normal( μ = 0, σ = 1). You can then use the standard normal cdf tables to get probabilities.

If you are looking at the mean of a sample, then remember that for any sample with a large enough sample size the mean will be normally distributed. This is called the Central Limit Theorem.

If a sample of size is is drawn from a population with mean μ and standard deviation σ then the sample average xBar is normally distributed

with mean μ and standard deviation σ /√(n)

An applet for finding the values
http://www-stat.stanford.edu/~naras/jsm/FindProbability.html

calculator
http://stattrek.com/Tables/normal.aspx

how to read the tables
http://rlbroderson.tripod.com/statistics/norm_prob_dist_ed9.html

In this question we have
X ~ Normal( μx = 8 , σx² = 9 )
X ~ Normal( μx = 8 , σx = 3 )

Find P( 8 < X < 14 )
= P( ( 8 – 8 ) / 3 < ( X – μ ) / σ < ( 14 – 8 ) / 3 )
= P( 0 < Z < 2 )
= P( Z < 2 ) – P( Z < 0 )
= 0.9772499 – 0.5
= 0.4772499

Find P( 5 < X < 11 )
= P( ( 5 – 8 ) / 3 < ( X – μ ) / σ < ( 11 – 8 ) / 3 )
= P( -1 < Z < 1 )
= P( Z < 1 ) – P( Z < -1 )
= 0.8413447 – 0.1586553
= 0.6826895

0.6826895 * 2000 = 1365.379 patients have clotting time between 5 and 11 seconds.

Find P( X > 11 )
P( ( X – μ ) / σ > ( 11 – 8 ) / 3 )
= P( Z > 1 )
= P( Z < -1 )
= 0.1586553

This entry was posted on Friday, November 27th, 2009 at 11:07 am and is filed under Blood Clot Patients. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.