Finding Probabilities for Normal Distributions In Exercises 7–12, find the indicated probabilities. If convenient, use technology to find the probabilities.
         
Health Club Schedule The amounts of time per workout an athlete uses a stairclimber are normally distributed, with a mean of 20 minutes and a standard deviation of 5 minutes. Find the probability that a randomly selected athlete uses a stairclimber for (a) less than 17 minutes.

Verified step by step guidance

1. Step 1: Identify the key parameters of the normal distribution. The problem states that the mean (μ) is 20 minutes and the standard deviation (σ) is 5 minutes. The random variable X represents the time an athlete uses the stairclimber.
2. Step 2: Standardize the value of 17 minutes to a z-score using the z-score formula: z = (X - μ) / σ. Substitute X = 17, μ = 20, and σ = 5 into the formula.
3. Step 3: After calculating the z-score, use the standard normal distribution table (or technology) to find the cumulative probability corresponding to the calculated z-score. This cumulative probability represents the probability that the time is less than 17 minutes.
4. Step 4: Interpret the cumulative probability. Since the problem asks for the probability of 'less than 17 minutes,' the cumulative probability directly gives the answer.
5. Step 5: If using technology, input the mean, standard deviation, and the value of 17 into a statistical software or calculator to find the probability directly. Ensure the settings are for a normal distribution.

Key Concepts

Normal Distribution

Normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, the time spent on a stairclimber follows this distribution, indicating that most athletes will use it around the mean time, with fewer athletes using it for significantly shorter or longer durations.

Mean and Standard Deviation

The mean is the average value of a dataset, while the standard deviation measures the amount of variation or dispersion from the mean. In this scenario, the mean workout time is 20 minutes, and the standard deviation of 5 minutes indicates how much individual workout times typically deviate from this average.

Z-Scores

A Z-score represents the number of standard deviations a data point is from the mean. It is calculated by subtracting the mean from the value and dividing by the standard deviation. For this problem, calculating the Z-score for 17 minutes will help determine the probability of an athlete using the stairclimber for less than that time by referencing standard normal distribution tables or software.