Question

The amounts of time per workout an athlete uses a stair climber are normally distributed, with a mean of 26 minutes and a standard deviation of 6 minutes. Find the probability that a randomly selected athlete uses a stair climber for (a) less than 20 minutes, (b) between 25 and 34 minutes, and (c) more than 40 minutes.

Answer 1

This question requires calculating probabilities for different time intervals using the Z-score method within a normal distribution, including looking up values in a standard normal distribution table.

Step-by-step explanation:

The question involves using the properties of the normal distribution to calculate probabilities for the time spent using a stair climber by an athlete.

1. To find the probability that a randomly selected athlete uses the stair climber for less than 20 minutes, we can use the Z-score formula: Z = (X - μ) / σ, where X is the value of interest (20 minutes), μ is the mean (26 minutes), and σ is the standard deviation (6 minutes). Once the Z-score is calculated, we look up this value in the standard normal distribution table.
2. For the probability between 25 and 34 minutes, we find the Z-scores for both values and look up the corresponding probabilities, then subtract the smaller probability from the larger one.
3. To find the probability that an athlete uses the stair climber for more than 40 minutes, we calculate the Z-score for 40, find its corresponding probability, and then subtract this value from 1, as we are looking for the probability above this Z-score.