"In Problems 5–14, a discrete random variable is given. Assume the probability of the random variable will be approximated using the normal distribution. Describe the area under the normal curve that will be computed. For example, if we wish to compute the probability of finding at least five defective items in a shipment, we would approximate the probability by computing the area under the normal curve to the right of x = 4.5.
The probability that the number of people with blood type O-negative is between 18 and 24, inclusive"

Verified step by step guidance

1. Identify the discrete random variable and the range of interest. Here, the variable is the number of people with blood type O-negative, and we want the probability that this number is between 18 and 24, inclusive.
2. Since the problem involves approximating a discrete distribution with a normal distribution, apply the continuity correction. For the interval 18 to 24 inclusive, adjust the bounds to 17.5 and 24.5 to better approximate the discrete probabilities.
3. Express the probability in terms of the normal distribution as the area under the curve between the corrected bounds. This means we want to find the area under the normal curve from \(x = 17.5\) to \(x = 24.5\).
4. To compute this area, convert the bounds to their corresponding z-scores using the formula \(z = \frac{x - \mu}{\sigma}\), where \(\mu\) is the mean and \(\sigma\) is the standard deviation of the approximating normal distribution.
5. Finally, use the standard normal distribution table or technology to find the probabilities corresponding to these z-scores, and subtract to find the area between them, which approximates the desired probability.

Key Concepts

Discrete to Continuous Approximation

When a discrete random variable is approximated by a continuous distribution like the normal, a continuity correction is applied. This involves adjusting the discrete values by 0.5 to better estimate probabilities, such as using 17.5 and 24.5 to approximate the probability between 18 and 24 inclusive.

Normal Distribution and Area Under the Curve

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve. Probabilities correspond to areas under this curve between specified values, representing the likelihood that a random variable falls within that range.

Probability Calculation Using the Normal Approximation

To find the probability of a discrete event using the normal approximation, convert the discrete bounds with continuity correction, then calculate the area under the normal curve between these adjusted values. This area represents the approximate probability of the event occurring.