The number of birth defects in a region is commonly modeled as having a binomial distribution, with a rate of four birth defects per 100 births considered a typical rate in the United States. What is the probability that a county that had 200 births during the year would have between 4 and 12 birth defects. Use both the binomial distribution and the Normal approximation with the appropriate continuity correction from the table below. Is the normal approximation reasonable?
The number of birth defects in a region is commonly modeled as having a binomial distribution, with a rate of four birth defects per 100 births considered a typical rate in the United States. What is the probability that a county that had 200 births during the year would have between 4 and 12 birth defects. Use both the binomial distribution and the Normal approximation with the appropriate continuity correction from the table below. Is the normal approximation reasonable?
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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The number of birth defects in a region is commonly modeled as having a binomial distribution, with a rate of four birth defects per 100 births considered a typical rate in the United States. What is the probability that a county that had 200 births during the year would have between 4 and 12 birth defects. Use both the binomial distribution and the Normal approximation with the appropriate continuity correction from the table below. Is the normal approximation reasonable?
Transcribed Image Text:### Binomial Distribution and Normal Approximation
This table provides a comparison between binomial distributions and their normal approximations. It includes conditions and notes about the inclusion of certain values.
#### Columns:
1. **Binomial Distribution:**
- \( P(x = c) \) : Probability where \( x = 10 \)
- \( P(x > c) \) : Probability where \( x > 10 \)
- \( P(x \leq c) \) : Probability where \( x \leq 10 \)
- \( P(x < c) \) : Probability where \( x < 10 \)
- \( P(x \geq c) \) : Probability where \( x \geq 10 \)
- \( P(a < x < b) \) : Probability where \( 9 < x < 11 \)
2. **Normal Approximation:**
- \( P(c - 0.5 < x < c + 0.5) \) : Represents the probability \( (9.5 < x < 10.5) \)
- \( P(x > c + 0.5) \) : Represents the probability \( (x > 10.5) \)
- \( P(x < c + 0.5) \) : Represents the probability \( (x < 10.5) \)
- \( P(x < c - 0.5) \) : Represents the probability \( (x < 9.5) \)
- \( P(x > c - 0.5) \) : Represents the probability \( (x > 9.5) \)
- \( P(a - 0.5 < x < b + 0.5) \) : Represents the probability \( (8.5 < x < 11.5) \)
3. **Notes:**
- For \( P(x = c) \) and \( P(x \leq c) \) and \( P(x \geq c) \), the value \( c \) is included.
- For \( P(x > c) \) and \( P(x < c) \), the value \( c \) is not included.
This table is useful for understanding how binomial probabilities can be approximated using a normal distribution, highlighting adjustments for continuity.
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