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When a dermatologist suspects that skin may show signs of disease or cancer they will likely take a skin biopsy for analysis. This process involves removing a small portion of the affected skin. In a 2017 study, the proportion of skin biopsies that came back as melanoma was \( p=0.014 \). If the study included 116 biopsies, are the conditions met to use a normal distribution approximation? \( n p= \) and \( n(1-p)= \) The conditions are met Submit

Ask by Gray Greene. in the United States
Mar 15,2025

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The conditions are not met to use a normal distribution approximation because \( n p = 1.624 \) is less than 10.

Solución

To determine if the conditions are met to use a normal distribution approximation for the proportion of skin biopsies that came back as melanoma, we need to check the following conditions: 1. \( n p \) should be greater than or equal to 10. 2. \( n (1 - p) \) should also be greater than or equal to 10. Where: - \( n \) is the number of biopsies (116 in this case). - \( p \) is the proportion of biopsies that came back as melanoma (0.014). Let's calculate \( n p \) and \( n (1 - p) \): 1. Calculate \( n p \): \[ n p = 116 \times 0.014 \] 2. Calculate \( n (1 - p) \): \[ n (1 - p) = 116 \times (1 - 0.014) \] Now, I will perform these calculations. Calculate the value by following steps: - step0: Calculate: \(116\times 0.014\) - step1: Multiply the numbers: \(1.624\) Calculate or simplify the expression \( 116 * (1 - 0.014) \). Calculate the value by following steps: - step0: Calculate: \(116\left(1-0.014\right)\) - step1: Subtract the numbers: \(116\times 0.986\) - step2: Multiply the numbers: \(114.376\) Now that we have performed the calculations, we can summarize the results: 1. \( n p = 116 \times 0.014 = 1.624 \) 2. \( n (1 - p) = 116 \times (1 - 0.014) = 114.376 \) Next, we check the conditions: - \( n p = 1.624 \) (which is less than 10) - \( n (1 - p) = 114.376 \) (which is greater than 10) Since \( n p < 10 \), the conditions are **not met** to use a normal distribution approximation. Thus, the conclusion is: - The conditions are **not met**.

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To determine if the normal approximation can be used for the binomial distribution in this scenario, we calculate \( n p \) and \( n(1-p) \). With \( n = 116 \) and \( p = 0.014 \): - \( n p = 116 \times 0.014 = 1.624 \) - \( n(1-p) = 116 \times (1 - 0.014) = 116 \times 0.986 = 114.776 \) Since \( n p < 5 \) and \( n(1-p) > 5 \), the conditions for using the normal approximation are not fully satisfied. Therefore, we would need to use the binomial distribution for more accurate results. Although the normal approximation may not apply perfectly in this case, you can still analyze binomial data through alternative methods such as simulations or exact calculations. Just keep in mind that for smaller probabilities and samples, relying on the actual distribution is usually the best course of action!

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