"for 2000 patients, blood-clotting time was normally distributed with a mean of 8 seconds and a standard deviation of 3 seconds. What percent had blood-clotting times between 5 and 11 seconds

You usually need an error function table or computer tool to do this kind of problem. I am a bit surprised they teach this in algebra 2.

I like to use the JAVA tool at
http://psych.colorado.edu/~mcclella/java/normal/accurateNormal.html

The answer in this case is 68.3%

That will always be the answer when you are talking about the region from
one standard deviation below the mean to one standard deviation above the mean, as is the case here.

To determine the percentage of patients with blood-clotting times between 5 and 11 seconds, we need to calculate the area under the normal distribution curve within this range. Here's how you can do it:

Step 1: Calculate the Z-scores for the lower and upper limits.

First, we need to convert the given values of 5 seconds and 11 seconds into Z-scores to use the standard normal distribution table.

Z-score = (X - μ) / σ

For the lower limit (X = 5 seconds):
Z1 = (5 - 8) / 3 = -1

For the upper limit (X = 11 seconds):
Z2 = (11 - 8) / 3 = 1

Step 2: Find the area under the curve.

Using the Z-scores obtained in step 1, we can find the respective areas under the curve using a standard normal distribution table or a calculator.

You can use a standard normal distribution table to find the area between Z1 and Z2. Look up the Z-scores in the table and subtract the values to find the area.

Area = Φ(Z2) - Φ(Z1)

Where Φ is the cumulative distribution function (CDF) of the standard normal distribution.

For Z1 = -1, Φ(-1) = 0.1587
For Z2 = 1, Φ(1) = 0.8413

Area = 0.8413 - 0.1587 = 0.6826

Step 3: Convert the area to a percentage.

The area calculated in step 2 represents the percentage of the data that falls between the two limits, 5 and 11 seconds.

Percentage = Area * 100

Percentage = 0.6826 * 100 = 68.26%

Therefore, approximately 68.26% of the patients had blood-clotting times between 5 and 11 seconds.