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Chapter 6: Problem 10

The values of \(n p\) and \(n q\) must equal at least what value for the normal approximation to be used as an estimate for binomial probabilities?

Short Answer

Expert verified
The values of \( np \) and \( nq \) must each be at least 5.

Step by step solution

01

Understand the Normal Approximation Requirement

The normal approximation to the binomial distribution can be used when certain conditions are satisfied. These conditions relate to the product of the number of trials \( n \) and the probabilities \( p \) and \( q \). Both \( np \) and \( nq \) must meet a minimum threshold for the approximation to be reliable.
02

Determine the Minimum Value

The general rule of thumb is that the values of \( np \) and \( nq \) should both be greater than or equal to 5. This ensures that the binomial distribution is close enough to the normal distribution for the approximation to be valid.
03

Conclusion

Therefore, for the normal approximation to be appropriately used for estimating binomial probabilities, both \( np \) and \( nq \) must be at least 5.

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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 fundamental concept in statistics, used to model scenarios where there are two possible outcomes in each trial. These scenarios are known as 'Bernoulli trials', named after the Swiss mathematician Jacob Bernoulli. Think of simple examples such as flipping a coin, where the outcomes are heads or tails:

  • Independent Trials: Each trial has no influence on the outcome of another.
  • Fixed Number of Trials: The process is repeated a specific number of times, denoted as \( n \).
  • Two Possible Outcomes: Typically labeled 'success' or 'failure', often translated into probabilities \( p \) and \( q \) respectively, where \( q = 1 - p \).
A central aspect of the binomial distribution is its probability mass function, which calculates the probability of achieving a certain number of successes in \( n \) trials. It's given by the formula \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( \binom{n}{k} \) is the binomial coefficient and \( k \) is the number of successes.
Probability Thresholds
In order to use the normal approximation for binomial distributions, particular probability thresholds must be achieved. These are numerical conditions on the variables involved, specifically \( np \) and \( nq \):

  • Threshold Requirement: Both \( np \) and \( nq \) must be equal to or greater than 5.
  • Ensures Accuracy: This threshold helps ensure that the binomial distribution sufficiently resembles a normal distribution.
Meeting these thresholds is crucial because the shape of a binomial distribution becomes more symmetric and bell-shaped, similar to a normal distribution, when \( np \) and \( nq \) are large enough. This approximation works well when the probability of success \( p \) isn't too close to 0 or 1, which might otherwise skew the distribution significantly. In these cases, despite having large sample sizes, the normal approximation might not hold.
Statistical Conditions
Statistical conditions refer to the set of criteria or prerequisites that must be fulfilled for a particular statistical method to be applied correctly. In the context of approximating a binomial distribution using a normal distribution, conditions ensure the method鈥檚 reliability:

  • Sufficiently Large Sample Size: Ensures that the law of large numbers applies, making the distribution more predictable.
  • Symmetrical Shape: When achieved, this allows the normal approximation to provide a close estimate, reducing potential errors in calculations.
  • Central Limit Theorem (CLT): States that, with a large enough sample size, the sampling distribution of the sample mean will tend to be normal, helping form the foundation for this approximation.
These statistical conditions highlight why thresholds for \( np \) and \( nq \) are integral, as they provide the structural integrity needed for the approximation to a bell-shaped curve, characteristic of normal distributions.

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Most popular questions from this chapter

A normal distribution has a mean equal to 10 . What is the standard deviation of this normal distribution if the cutoff for the top \(5 \%\) is \(x=12.47\) ?

State whether the first area is bigger, the second area is bigger, or the two areas are equal in each of the following situations: a. The area to the left of \(z=1.00\) and the area to the right of \(z=-1.00\) b. The area to the right of \(z=1.00\) and the area to the right of \(z=-1.00\) c. The area between the mean and \(z=1.20\) and the area to the right of \(z=0.80\) d. The area to the left of the mean and the area between \(z=\pm 1.00\) e. The area to the right of \(z=1.65\) and the area to the left of \(z=-1.65\)

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