"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 exactly eight defective parts are in the shipment
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Verified step by step guidance

1. Identify the discrete random variable of interest, which in this case is the number of defective parts in the shipment, and note that we want the probability that exactly 8 defective parts occur.
2. Since the variable is discrete, apply the continuity correction to approximate the probability using the normal distribution. For "exactly 8", consider the interval from 7.5 to 8.5 to capture the probability mass at 8.
3. Determine the mean (\$\mu\$) and standard deviation (\$\sigma\$) of the discrete random variable, which are necessary parameters for the normal approximation. These should be given or calculated from the problem context.
4. Convert the interval bounds 7.5 and 8.5 to their corresponding z-scores using the formula: \[z = \frac{x - \mu}{\sigma}\] where \$x\$ is the boundary value (7.5 or 8.5).
5. The probability that exactly 8 defective parts occur is approximated by the area under the normal curve between the two z-scores calculated. This corresponds to the probability \$P(7.5 < X < 8.5)\$ under the normal distribution.

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 value by ±0.5 to better estimate probabilities, since the normal distribution is continuous and the discrete variable takes integer values.

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 over specific intervals. Calculating the probability of an event involves finding the area under the curve between or beyond certain points.

Probability of Exact Values Using Normal Approximation

To find the probability that a discrete variable equals a specific value using the normal approximation, compute the area under the normal curve between the continuity-corrected bounds (value - 0.5 and value + 0.5). This area approximates the probability of the exact discrete outcome.