Assume that the probability of the binomial random variable will be approximated using the normal distribution. Describe the area under the normal curve that will be computed.Find the probability that there are exactly 12 defective parts in a shipment.
When a binomial random variable has a large number of trials and a moderate probability of success, we can approximate its probability distribution using the normal distribution
When a binomial random variable has a large number of trials and a moderate probability of success, we can approximate its probability distribution using the normal distribution.
To find the probability that there are exactly 12 defective parts in a shipment, we need to use the normal approximation to the binomial distribution.
First, we need to calculate the mean (μ) and standard deviation (σ) of the binomial random variable. The mean of a binomial random variable is given by μ = np, where n is the number of trials and p is the probability of success. The standard deviation is given by σ = √(np(1-p)).
Next, we use these values to approximate the binomial probability as a normal distribution. In this case, we are interested in finding the probability of exactly 12 defective parts, which can be represented as P(X = 12), where X is the binomial random variable.
To use the normal approximation, we adjust the values slightly by adding or subtracting 0.5 to the discrete value to account for continuity. So, we are essentially finding P(11.5 < X < 12.5). Now, we transform the binomial distribution into a standard normal distribution using the formula: Z = (X - μ) / σ, where Z is the standard normal random variable. In this case, Z = (12 - μ) / σ. Once we have calculated the Z-score, we can use a standard normal distribution table or a calculator to find the area under the normal curve. The area represents the probability that falls within the range 11.5 to 12.5. Note that in the normal distribution, we are calculating an area under the curve, which implies a probability. This area represents the probability of having exactly 12 defective parts in the shipment. It is important to remember that the normal approximation is an estimation and may not be completely accurate, especially if the number of trials is not large or if the probability of success is close to 0 or 1.
More Answers:
Analyzing Histogram Characteristics to Determine If a Normal Distribution Model Fits the DataObservational Study | Association between TV in Bedroom and Higher BMI in Adolescents, with Potential Lurking Variables
Comparing Exam Scores | ACT vs. SAT – Determining Your Performance Relative to Mean and Standard Deviation