Question

Find the normal approximation to \(P(355 \leq x \leq 360)\) for a binomial probability distribution with \(n=400\) and \(p=.9\).

Short answer

Question: Using the normal approximation method, find the approximate probability of observing between 355 and 360 successes in a binomial experiment with n=400 trials and p=0.9. Answer: The normal approximation to the probability of observing between 355 and 360 successes is approximately 0.2977.

Step-by-step solution

Find the mean and standard deviation of the binomial distribution

We are given the parameters of a binomial distribution: number of trials (n=400) and probability of success (p=0.9). Recall that the mean (μ) and standard deviation (σ) of a binomial distribution are given by the following formulas: Mean: \(μ = n * p\) Standard deviation: \(σ = \sqrt{n * p * (1-p)}\) Using the given values, we get: Mean: \(μ = 400 * 0.9 = 360\) Standard deviation: \(σ = \sqrt{400 * 0.9 * (1-0.9)} = \sqrt{400 * 0.9 * 0.1} = 6\)

Calculate the z-scores corresponding to the given range

Now we need to find the z-scores corresponding to the given range (355 to 360). The z-score formula is as follows: \(z = \frac{X - μ}{σ}\) Using the mean and standard deviation calculated in step 1, we get: \(z_{355} = \frac{355 - 360}{6} = -\frac{5}{6} \approx -0.833\) \(z_{360} = \frac{360 - 360}{6} = 0\)

Find the probability using the z-scores

Now that we have the z-scores, we can use a standard normal distribution table or calculator to find the probability of observing a value within this range. Using a standard normal distribution table (or online calculator), we find the probability for each of the z-scores: \(P(z \leq -0.833) \approx 0.2023\) (probability of observing a z-score less than or equal to -0.833) \(P(z \leq 0) = 0.5\) (probability of observing a z-score less than or equal to 0) Now, we need to subtract these probabilities to find the probability of the range in question: \(P(355 \leq x \leq 360) \approx P(z \leq 0) - P(z \leq -0.833) \approx 0.5 - 0.2023 = 0.2977\) Therefore, the normal approximation to the probability \(P(355 \leq x \leq 360)\) is approximately 0.2977.

Key concepts

Binomial Distribution

A binomial distribution is a common type of probability distribution. It describes the number of successes in a fixed number of binary trials, where each trial has two possible outcomes, often termed as "success" and "failure." Let's break it down further:

• **Number of Trials ("):** In our example, there are 400 trials.
• **Probability of Success (\(p\)):** Each trial has a 0.9 probability of success.

This distribution is helpful in real-world scenarios where the outcome is binary, like a flipped coin landing heads or tails, or a test showing positive or negative. It is defined by two parameters: the number of trials and the probability of success in each trial. Understanding these parameters helps us compute further statistical measurements, such as the mean and standard deviation, which lead to approximations using other distributions.

Z-scores

The concept of a z-score is pivotal when dealing with normal distributions. A z-score measures how many standard deviations an element is from the mean. The formula is:

\[ z = \frac{X - \mu}{\sigma} \]

• **\(X\)** represents a data point.
• **\(\mu\)** is the mean of the dataset.
• **\(\sigma\)** is the standard deviation.

Z-scores allow us to compare data points from different distributions. By converting a data point to a z-score, we're essentially converting the data into a standard normal distribution with mean 0 and variance 1. This transformation helps determine how typical or atypical a particular data point is within its dataset. In the solved example, z-scores are used to help identify probabilities in a standard normal distribution table, bridging a vast range of data points with standardized z-units.

Standard Deviation

Standard deviation is a statistic that measures the dispersion or spread of a dataset relative to its mean. It tells us how much the values in a dataset deviate from the mean value.

In the context of a binomial distribution, standard deviation is calculated using the formula:
\[ \sigma = \sqrt{n \times p \times (1-p)} \]
Here's what the pieces mean:

• **\(n\)** is the number of trials.
• **\(p\)** is the probability of success.
• **\(1-p\)** is the probability of failure.

In practical terms, a small standard deviation means the data points are close to the mean, while a large standard deviation indicates a wide spread around the mean. In our example, the standard deviation is used to compute z-scores, giving insight into the likelihood of observing a particular range of successes over trials.

Mean

The mean of a dataset is the sum of all the data points divided by the total number of observations. It is commonly referred to as the average, providing a central value for data points.

For a binomial distribution, the mean is calculated as follows:
\[ \mu = n \times p \]

• **\(n\)** is the number of trials.
• **\(p\)** is the probability of success per trial.

Knowing the mean of a binomial distribution helps in understanding the expected number of successes in a series of trials. In the example exercise, the mean is critical in the conversion of raw scores to z-scores, which subsequently leads to the normal approximation and probability calculation of the specified range. This centrality makes the mean a fundamental building block in probability and statistics, bridging theoretical distribution to practical approximation.