Normal Approximation to the Binomial Distribution

Introduction

The normal approximation to the binomial distribution is a useful technique for estimating binomial probabilities when the number of trials is large and the probability of success is not too close to 0 or 1. This method allows us to use the normal distribution, which is continuous, to approximate the probabilities of a discrete binomial distribution.

Conditions for Normal Approximation

• Applicability: The binomial distribution can be approximated by a normal distribution if both and , where is the number of trials, is the probability of success, and is the probability of failure.

• Parameters:

  • Mean:

  • Standard deviation:

Steps for Using the Normal Approximation

1. Verify the binomial distribution applies: Specify , , and .

2. Check the normal approximation conditions: Ensure and .

3. Find the mean and standard deviation:

4. Apply the continuity correction: Since the binomial distribution is discrete and the normal distribution is continuous, add or subtract 0.5 to the value of as appropriate.

5. Find the corresponding z-score(s):

   • Where is the value of after applying the continuity correction.

6. Find the probability: Use the standard normal table to find the probability corresponding to the calculated z-score(s).

Continuity Correction

When approximating a binomial probability with a normal distribution, a continuity correction is necessary because the binomial distribution is discrete and the normal distribution is continuous. This correction involves adjusting the discrete value by 0.5 units:

• For exactly c:

• For at most c:

• For fewer than c:

• For at least c:

• For more than c:

This adjustment ensures that the probability calculated using the normal distribution more accurately reflects the probability from the binomial distribution.

Table: Binomial to Normal Probability Conversion

Example 1: Using a Continuity Correction

• Problem: Find the probability of getting at least 158 successes.

• Solution:

  • The discrete midpoint values are 158, 159, 160, ...

  • The corresponding interval for the continuous normal distribution is .

  • The normal distribution probability is .

Example 2: Approximating a Binomial Probability

• Problem: In a survey of 8- to 18-year-old heavy media users in the United States, 47% said they get fair or poor grades (C and below). You randomly select 45 such users and ask them whether they get fair or poor grades. What is the probability that fewer than 20 of them respond yes?

• Solution:

  • Given: , ,

  • Calculate mean and standard deviation:

  • Apply the continuity correction: "Fewer than 20" becomes .

  • Find the z-score:

  • Find the probability using the standard normal table:

  • Conclusion: The probability that fewer than twenty 8- to 18-year-olds respond yes is approximately 0.3121, or about 31.21%.

Summary Table: Steps for Normal Approximation

Key Points

• The normal approximation is most accurate when is large and is not too close to 0 or 1.

• Continuity correction is essential for accurate approximation.

• Always check the conditions before applying the normal approximation.

Additional info: The continuity correction bridges the gap between discrete and continuous probability models, ensuring that the area under the normal curve closely matches the probability from the binomial distribution.