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Magazine Chapter 6: Problem 23
Recall that for use of a normal distribution as an approximation to the binomial distribution, the conditions \(n p \geq 5\) and \(n q \geq 5\) must be met. For each given probability, compute the minimum sample size needed for use of the normal approximation. a. \(p=0.1\) d. \(p=0.8\) b. \(p=0.3\) e. \(p=0.9\) c. \(p=0.5\)
Short Answer
Expert verified
a. 50, b. 17, c. 10, d. 25, e. 50.
Step by step solution
01
Introduction to Minimum Sample Size Calculation
To use the normal distribution to approximate the binomial distribution, we need to ensure that both conditions \(np \geq 5\) and \(nq \geq 5\) are satisfied, where \(q = 1 - p\). The goal is to find the minimum value of \(n\) that meets these criteria for each value of \(p\).
02
Calculate for p=0.1
Given \(p = 0.1\), the condition \(np \geq 5\) translates to \(0.1n \geq 5\), leading to \(n \geq \frac{5}{0.1} = 50\). For \(nq \geq 5\), where \(q = 0.9\), the condition becomes \(0.9n \geq 5\), resulting in \(n \geq \frac{5}{0.9} \approx 5.56 \). Both conditions give \(n \geq 50\) since 50 satisfies both.
03
Calculate for p=0.3
With \(p = 0.3\), \(np \geq 5\) becomes \(0.3n \geq 5\), so \(n \geq \frac{5}{0.3} \approx 16.67\). For \(nq \geq 5\), where \(q = 0.7\), the inequality \(0.7n \geq 5\) gives \(n \geq \frac{5}{0.7} \approx 7.14\). The minimum \(n\) that satisfies both is \(n \geq 17\).
04
Calculate for p=0.5
For \(p = 0.5\), both conditions become the same: \(0.5n \geq 5\). Solving this gives \(n \geq \frac{5}{0.5} = 10\). Thus, the minimum \(n\) needed is 10.
05
Calculate for p=0.8
When \(p = 0.8\), \(np \geq 5\) translates to \(0.8n \geq 5\), which gives \(n \geq \frac{5}{0.8} = 6.25\). With \(q = 0.2\), \(0.2n \geq 5\) results in \(n \geq \frac{5}{0.2} = 25\). Therefore, the minimum \(n\) satisfying both conditions is \(n \geq 25\).
06
Calculate for p=0.9
For \(p = 0.9\), \(np \geq 5\) yields \(0.9n \geq 5\), hence \(n \geq \frac{5}{0.9} \approx 5.56\). With \(q = 0.1\), \(0.1n \geq 5\) gives \(n \geq \frac{5}{0.1} = 50\). The minimum \(n\) that meets both criteria is \(n \geq 50\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Distribution
A binomial distribution is a common probability distribution used in statistics to model the number of successes in a fixed number of trials. Each trial is independent, meaning the outcome of one trial does not affect another, and there are only two possible outcomes: success or failure. It is defined by two parameters: the number of trials, denoted as \( n \), and the probability of success in a single trial, denoted as \( p \). The probability of failure is noted as \( q = 1 - p \). When analyzing a scenario such as flipping a coin multiple times or testing batch quality in a factory, the binomial distribution helps us understand the likelihood of achieving a specific number of successes. This distribution is foundational for various statistical measures and calculations, helping to predict outcomes and make informed decisions.
Minimum Sample Size
To use the normal approximation of a binomial distribution, certain conditions must be met to ensure accuracy. One key requirement is determining the minimum sample size \( n \). This ensures the distribution can be appropriately approximated by a normal distribution. The conditions require that both \( np \geq 5 \) and \( nq \geq 5 \) are satisfied, where \( q = 1 - p \). These inequalities ensure that the distribution of outcomes is sufficiently spread to resemble a normal distribution, allowing for smoother and more reliable approximations. For example, if \( p = 0.1 \), \( n \) must be at least 50, meeting both conditions. Calculating the minimum sample size is crucial in designing experiments and studies, ensuring the collected data is valid and can justify statistical approximations.
Probability Conditions
Probability conditions play a crucial role in the normal approximation of the binomial distribution. These conditions ensure that the approximation remains valid for statistical inference. When using the normal approximation, the two main conditions are:
- \( np \geq 5 \) - the expected number of successes should be at least 5.
- \( nq \geq 5 \) - the expected number of failures should also be at least 5.
Statistical Calculations
Statistical calculations often involve approximating more complex distributions with simpler ones. The normal approximation to a binomial distribution is a method that simplifies calculations and analysis when dealing with large sample sizes.
Once the minimum sample size conditions are satisfied, you can perform various statistical analyses like calculating probabilities, determining confidence intervals, or estimating distributions.
For instance, the central limit theorem supports this approximation, indicating that as the sample size grows, the distribution of the sample mean will approach normality regardless of the original distribution.
This approximation method streamlines the computation process in statistics, providing an efficient approach for handling larger datasets and more intricate probability problems.
