Health Club Schedule

The time per workout an athlete uses a stairclimberis normally distributed with a mean of 20 mins and a standard deviation of 5 min. An athlete is randomly selected. Find the probability that the athlete uses a stairclimber between 20 and 28 mins.

Z = (score-mean)/SD

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

To find the probability that the athlete uses a stair climber between 20 and 28 minutes, we can use the concept of the standard normal distribution.

Step 1: Standardize the values
To apply the standard normal distribution, we need to convert the given values into z-scores. The z-score formula is:

z = (X - μ) / σ

where X is the given value, μ is the mean, and σ is the standard deviation.

For the lower value of 20 minutes:
z1 = (20 - 20) / 5 = 0

For the upper value of 28 minutes:
z2 = (28 - 20) / 5 = 1.6

Step 2: Find the probabilities
Now that we have the z-scores, we can use the standard normal distribution table or a statistical calculator to find the probabilities associated with these z-scores.

Looking up the z-scores in a standard normal distribution table, we find:

For z = 0, the probability is 0.5000
For z = 1.6, the probability is 0.9452

Step 3: Calculate the probability between the two values
To find the probability between the two values, we subtract the probability associated with the lower value from the probability associated with the upper value:

P(20 ≤ X ≤ 28) = P(X ≤ 28) - P(X ≤ 20)
= 0.9452 - 0.5000
= 0.4452

Therefore, the probability that the athlete uses a stair climber between 20 and 28 minutes is approximately 0.4452, or 44.52%.