recent 10K run the finisher are norm dist with a mean of 55 min and stand dev of 13

A) determine the percentage of finishers with times between 45 and 75 min
B) determine the percentage of finshers less than 80 min
C) obtain the 30th percentile
D) find the 8th decile

Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores.

Reverse the process for the last 2 problems.

30th percentile = lowest 30%

8th decile = lowest 80%

To solve these questions, we can use the properties of the normal distribution. To find the required values, we will need to standardize the values (convert them into z-scores) using the formula:

z = (x - μ) / σ

Where:
- z is the z-score
- x is the score we want to standardize
- μ is the mean of the distribution
- σ is the standard deviation of the distribution

Let's calculate the values using this formula.

A) Percentage of finishers with times between 45 and 75 min:
1. Standardize the values:
z1 = (45 - 55) / 13 = -0.77
z2 = (75 - 55) / 13 = 1.54

2. Use a standard normal distribution table to find the percentage between these two z-scores. The table gives you the area to the left of the z-score. So you need to find the area for z2 and subtract the area for z1 from it.

Area between z1 and z2 = Area(z2) - Area(z1)
= Area(1.54) - Area(-0.77)

Using the table or a calculator, you can find the area for each z-score. Subtracting the smaller area from the larger one will give you the percentage of finishers with times between 45 and 75 min.

B) Percentage of finishers less than 80 min:
1. Standardize the value:
z = (80 - 55) / 13 = 1.92

2. Use a standard normal distribution table to find the percentage to the left of z = 1.92. This will give you the percentage of finishers with times less than 80 min.

C) 30th percentile:
1. Use the standard normal distribution table to find the z-score that corresponds to the 30th percentile. Look for the closest value to 0.30 in the table.

2. Use the formula to find the corresponding time:
x = μ + z * σ

Substitute the values of μ, σ, and the z-score you obtained from the table into the formula to get the 30th percentile time.

D) 8th decile:
1. The 8th decile corresponds to the 80th percentile. Use similar steps as in part C to find the z-score that corresponds to the 80th percentile.

2. Use the formula to find the corresponding time:
x = μ + z * σ

Substitute the values of μ, σ, and the z-score you obtained from the table into the formula to get the 8th decile time.