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Binomial distribution B(n,p) can be approximated to a normal distribution N(np, np(1−p)) if ____
  • a)
    n is large and p and 1-p are almost equal.
  • b)
    n is small and p is almost 1.
  • c)
    n is large and p is almost 0.
  • d)
    n is small and p and 1-p are almost equal.
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Binomial distributionB(n,p) can be approximated to a normal distributi...
Binomial distribution B(n,p) can be approximated to normal distribution N(np,np(1−p)) if n is large and p and 1-p are almost equal. Approximation generally improves as n increases (at least 20) and is better when p is not near to 0 or 1.
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Most Upvoted Answer
Binomial distributionB(n,p) can be approximated to a normal distributi...
Explanation:

Binomial Distribution vs. Normal Distribution:
- Binomial distribution B(n,p) is used to model the number of successes in a fixed number of trials, where each trial has the same probability of success p.
- Normal distribution N(np, np(1-p)) is a continuous probability distribution that is symmetric and bell-shaped.

Conditions for Approximation:
- When n is large and p and 1-p are almost equal, the binomial distribution can be approximated by a normal distribution.
- This is because when n is large, the binomial distribution becomes more symmetric and bell-shaped, resembling a normal distribution.

Reason for Option A:
- Option A states that n is large and p and 1-p are almost equal, which satisfies the conditions for approximating the binomial distribution to a normal distribution.
- In this scenario, the mean of the normal distribution is np and the standard deviation is sqrt(np(1-p)), which closely approximates the binomial distribution.

Conclusion:
- Therefore, option A is the correct choice as it correctly identifies the conditions under which the binomial distribution can be approximated by a normal distribution.
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