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 (b) between 20 and 28 minutes.

Verified step by step guidance

1. Step 1: Identify the key parameters of the normal distribution. The mean (μ) is 20 minutes, and the standard deviation (σ) is 5 minutes. The problem asks for the probability that the time spent on the stairclimber is between 20 and 28 minutes.
2. Step 2: Standardize the values 20 and 28 using the z-score formula: z = (x - μ) / σ. For x = 20, calculate z₁ = (20 - 20) / 5. For x = 28, calculate z₂ = (28 - 20) / 5.
3. Step 3: Use the standard normal distribution table or technology to find the cumulative probabilities corresponding to z₁ and z₂. The cumulative probability for z₁ represents the area under the curve to the left of z₁, and similarly for z₂.
4. Step 4: Subtract the cumulative probability for z₁ from the cumulative probability for z₂ to find the probability that the time spent is between 20 and 28 minutes. This is because the area between z₁ and z₂ represents the desired probability.
5. Step 5: Interpret the result. The final probability represents the likelihood that a randomly selected athlete spends between 20 and 28 minutes on the stairclimber, based on the given 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, it describes how the workout times of athletes are distributed around the average time of 20 minutes, with most athletes working out close to this mean and fewer athletes working out significantly longer or shorter.

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 of 20 minutes indicates the typical workout time, and the standard deviation of 5 minutes shows how much individual workout times vary from this average, helping to understand the spread of workout durations.

Finding Probabilities

Finding probabilities in a normal distribution involves calculating the area under the curve between two values, which can be done using z-scores or statistical software. For the given problem, we need to determine the probability that an athlete works out between 20 and 28 minutes, which requires integrating the normal distribution function or using a standard normal table to find the corresponding probabilities.