"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 at least 40 households have a gas stove"

Verified step by step guidance

1. Identify the discrete random variable and the event of interest. Here, the event is "at least 40 households have a gas stove," which means \( X \geq 40 \) where \( X \) is the number of households with a gas stove.
2. Since \( X \) is discrete, apply the continuity correction to approximate the probability using the normal distribution. For "at least 40," this means considering \( X \geq 39.5 \) in the continuous normal approximation.
3. Determine the mean \( \mu \) and standard deviation \( \sigma \) of the discrete random variable \( X \). These parameters are necessary to define the corresponding normal distribution \( N(\mu, \sigma^2) \).
4. Translate the problem into finding the area under the normal curve to the right of \( x = 39.5 \). This corresponds to \( P(X \geq 40) \approx P(Y \geq 39.5) \) where \( Y \sim N(\mu, \sigma^2) \).
5. Express the probability in terms of the standard normal variable \( Z \) by standardizing: \( Z = \frac{39.5 - \mu}{\sigma} \). Then, the probability is \( P(Z \geq \frac{39.5 - \mu}{\sigma}) \), which is the area under the standard normal curve to the right of this \( Z \)-value.

Key Concepts

Discrete to Continuous Approximation

This concept involves approximating a discrete probability distribution, like the binomial or Poisson, with a continuous distribution such as the normal distribution. It is useful when the sample size is large, allowing easier calculation of probabilities using the normal curve instead of summing discrete probabilities.

Continuity Correction

When approximating a discrete distribution with a normal distribution, a continuity correction adjusts for the difference between discrete points and continuous intervals. For example, to find P(X ≥ 40), we calculate the area to the right of 39.5 under the normal curve, improving the accuracy of the approximation.

Area Under the Normal Curve

The area under the normal curve represents probabilities for continuous random variables. To find the probability of an event, we calculate the area corresponding to the event's range on the curve, using z-scores and standard normal tables or software to determine the exact probability.