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Chapter 6: Problem 1

Binomial probability distributions depend on the number of trials \(n\) of a binomial experiment and the probability of success \(p\) on each trial. Under what conditions is it appropriate to use a normal approximation to the binomial?

Short Answer

Expert verified
Use normal approximation when \( n \) is large and both \( np \geq 10 \) and \( n(1-p) \geq 10 \).

Step by step solution

01

Define When a Normal Approximation is Appropriate

The normal approximation to the binomial distribution can generally be used when the number of trials \( n \) is large and the probability of success \( p \) is not too close to 0 or 1. This allows for the binomial distribution to be similar to a normal curve.
02

Utilize the Success-Failure Condition

For a normal approximation to a binomial distribution to be suitable, it is important to check the success-failure condition: both \( np \) and \( n(1-p) \) should be greater than or equal to 10. This ensures that the distribution is sufficiently symmetric to resemble a normal distribution.
03

Additional Notes on Approximations

If both \( np \) and \( n(1-p) \) are much greater than 10, the approximation becomes more accurate. If they are barely above 10, use the approximation cautiously, as it may not be precise.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Approximation
When working with binomial distributions, it's not always straightforward to calculate probabilities, especially as the number of trials increases. This is where the concept of normal approximation becomes useful. For a binomial distribution to be approximated by a normal distribution, certain conditions related to the number of trials and probability of success must be met.
  • Large Number of Trials: The condition requires that the number of trials \( n \) must be large. Although there's no strict rule for what 'large' means, generally when \(n\) is 30 or more, you can start to consider using a normal approximation.
  • Probability of Success: The probability \( p \) of success on each trial should not be too extreme (i.e., neither very close to 0 nor 1). This condition is vital because it ensures that the distribution isn't skewed.
With these conditions satisfied, the binomial distribution can simulate a symmetric, bell-shaped normal curve. It's a computationally easier alternative, making practical problem-solving more approachable.
Success-Failure Condition
The success-failure condition is crucial in deciding whether a normal approximation can be effectively applied to a binomial distribution. This condition relies on simple calculations using the number of trials \( n \) and the probability of success \( p \).
  • Calculate Successes: The product \( np \), which represents the expected number of successes, must be at least 10. This value indicates that there should be enough successes to form a symmetric shape.

  • Calculate Failures: Likewise, \( n(1-p) \), the expected number of failures, must also be at least 10. This ensures that there are enough occurrences of non-successes, contributing to the symmetry needed to reflect a normal distribution.
Both these criteria must be satisfied to reliably use normal approximation. If they are barely met, it's wise to proceed with care as the approximation might lack precision.
Probability of Success
The probability of success \( p \) in a binomial distribution is a fundamental component that impacts whether a normal approximation is suitable. It's the probability of the event you are interested in occurring in each trial of the experiment.
  • Ideal Range for \( p \): For a smooth normal curve, \( p \) should not be too close to 0 or 1. This helps keep the distribution from being too skewed, maintaining a balance between successes and failures.

  • Ensuring Balance: When \( p \) is about 0.5, the symmetrization is ideal, but as you move away from this value towards the extremes, the shape becomes skewed. Thus, achieving a balance is key for using normal approximation effectively.
Considering \( p \) in the context of establishing the success-failure condition is essential, as it affects both \( np \) and \( n(1-p) \). Always assess whether the conditions are met before opting for normal approximation based on \( p \).

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Most popular questions from this chapter

Templeton World is a mutual fund that invests in both U.S. and foreign markets. Let \(x\) be a random variable that represents the monthly percentage return for the Templeton World fund. Based on information from the Morningstar Guide to Mutual Funds (available in most libraries), \(x\) has mean \(\mu=1.6 \%\) and standard deviation \(\sigma=0.9 \%\) (a) Templeton World fund has over 250 stocks that combine together to give the overall monthly percentage return \(x .\) We can consider the monthly return of the stocks in the fund to be a sample from the population of monthly returns of all world stocks. Then we see that the overall monthly return \(x\) for Templeton World fund is itself an average return computed using all 250 stocks in the fund. Why would this indicate that \(x\) has an approximately normal distribution? Explain. Hint: See the discussion after Theorem 6.2. (b) After 6 months, what is the probability that the average monthly percentage return \(\bar{x}\) will be between \(1 \%\) and \(2 \% ?\) Hint: See Theorem \(6.1,\) and assume that \(x\) has a normal distribution as based on part (a). (c) After 2 years, what is the probability that \(\bar{x}\) will be between \(1 \%\) and \(2 \% ?\) (d) Compare your answers to parts (b) and (c). Did the probability increase as \(n\) (number of months) increased? Why would this happen? (e) Interpretation: If after 2 years the average monthly percentage return \(\bar{x}\) was less than \(1 \%,\) would that tend to shake your confidence in the statement that \(\mu=1.6 \% ?\) Might you suspect that \(\mu\) has slipped below \(1.6 \% ?\) Explain.

Sketch the areas under the standard normal curve over the indicated intervals and find the specified areas. To the right of \(z=-2.17\)

Find the indicated probability, and shade the corresponding area under the standard normal curve. $$P(-2.37 \leq z \leq 0)$$

Assume that \(x\) has a normal distribution with the specified mean and standard deviation. Find the indicated probabilities.$$P(x \geq 2) ; \mu=3 ; \sigma=0.25$$

Quick Start Company makes 12-volt car batteries. After many years of product testing, the company knows that the average life of a Quick Start battery is normally distributed, with a mean of 45 months and a standard deviation of 8 months. (a) If Quick Start guarantees a full refund on any battery that fails within the 36 -month period after purchase, what percentage of its batteries will the company expect to replace? (b) Inverse Normal Distribution If Quick Start does not want to make refunds for more than \(10 \%\) of its batteries under the full-refund guarantee policy, for how long should the company guarantee the batteries (to the nearest month)?
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