Consider a manufacturing process where 200 items are produced and each item has a 5% chance of being defective. Let \(X\) denote the number of defective items. Use the binomial distribution to model \(X\).
Proceed to approximate the probability \(P(X \leq 8)\) using the normal approximation with a continuity correction, and then evaluate the accuracy of this approximation in comparison to the exact binomial probability.
Question
For the given scenario, determine the parameters of the binomial distribution. Specifically, calculate the mean \(\mu\) and variance \(\sigma^2\) of \(X\), where \(X\) denotes the number of defective items.
Hint
Hint
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Use the formulas for the mean \(\mu = np\) and variance \(\sigma^2 = np(1-p)\) for a binomial distribution with parameters \(n\) and \(p\).
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Complete Answer
Mean \(\mu = 10\), Variance \(\sigma^2 = 9.5\)
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Question
Using the parameters from part (a), apply the normal approximation with a continuity correction to estimate \(P(X \leq 8)\). Show all relevant steps.
Hint
Hint
Hint
To approximate \(P(X \leq 8)\) using the normal distribution with continuity correction, adjust the discrete boundary and standardize using the mean and standard deviation from part (a). Remember to use \(X + 0.5\) for \(P(X \leq 8)\).
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Approximately \(P(X \leq 8) \approx 0.313\).
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Question
Critically evaluate the accuracy of the normal approximation by comparing it to the exact binomial probability for \(P(X \leq 8)\). Briefly explain any potential sources of error.
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Calculate the exact binomial probability for \(P(X \leq 8)\) using the binomial probability formula or a calculator. Compare this value with the normal approximation obtained in part (b) and consider the conditions under which normal approximation works best for binomial distributions.
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The normal approximation yields \(P(X \leq 8) \approx 0.313\), while the exact binomial probability is approximately 0.330. The absolute difference is approximately 0.017, indicating a reasonable approximation. Potential sources of error include approximating a discrete distribution with a continuous one and the inherent limitations of continuity correction, especially when \(p\) is not very close to 0.5 or when \(n\) is not extremely large.
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