Normal Approximation to the Binomial Distribution

Normal Probability Distributions

Normal Approximation to the Binomial Distribution

The normal distribution can be used to approximate the binomial distribution under certain conditions. This technique simplifies calculations for large sample sizes and is a key concept in inferential statistics.

• Binomial Distribution: A discrete probability distribution describing the number of successes in a fixed number of independent trials, each with the same probability of success.

• Normal Distribution: A continuous, symmetric, bell-shaped distribution characterized by its mean (μ) and standard deviation (σ).

Conditions for Normal Approximation

• The normal distribution can approximate the binomial distribution if both of the following are true:

  • n × p ≥ 5

  • n × q ≥ 5

• Where:

  • n = number of trials

  • p = probability of success

  • q = probability of failure (q = 1 - p)

Formulas for Mean and Standard Deviation

• Mean (μ):

  • $\mu = n \times p$

• Standard Deviation (σ):

  • $\sigma = \sqrt{n \times p \times q}$

Example

• Suppose 35% of teen drivers admit to texting while driving. In a random sample of 100 teen drivers (n = 100, p = 0.35):

  • n × p = 100 × 0.35 = 35 (≥ 5)

  • q = 1 - 0.35 = 0.65

  • n × q = 100 × 0.65 = 65 (≥ 5)

  • Therefore, the normal approximation is appropriate.

  • Mean: $\mu = 100 \times 0.35 = 35$

  • Standard deviation: $\sigma = \sqrt{100 \times 0.35 \times 0.65} \approx 4.77$

Continuity Correction

Because the binomial distribution is discrete and the normal distribution is continuous, a continuity correction is applied when approximating binomial probabilities with the normal distribution. This involves adjusting the endpoints of the interval by 0.5 units.

• Continuity Correction: Add or subtract 0.5 to the discrete x-value(s) to better approximate the area under the normal curve.

• For a single value x: Use the interval (x - 0.5, x + 0.5).

• For an interval [a, b] (inclusive): Use (a - 0.5, b + 0.5).

Examples of Continuity Correction

• Between 57 and 83 successes (inclusive):

  • Binomial interval: 57 ≤ x ≤ 83

  • Normal interval: 56.5 < x < 83.5

• At most 54 successes:

  • Binomial: x ≤ 54

  • Normal: x < 54.5

Steps for Using the Normal Approximation

1. Verify Binomial Conditions: Confirm the scenario fits a binomial setting (fixed n, independent trials, constant p).

2. Check Normal Approximation Conditions: Ensure n × p ≥ 5 and n × q ≥ 5.

3. Find Mean and Standard Deviation: Use the formulas above.

4. Apply Continuity Correction: Adjust the x-values as described.

5. Convert to z-score: Use the formula:

   • $z = \frac{x_{corr} - \mu}{\sigma}$

   • Where xcorr is the continuity-corrected value.

6. Find Probability: Use the standard normal table or technology to find the probability corresponding to the calculated z-score.

Worked Example

• 52% of drivers who use both alcohol and marijuana admit to driving aggressively. In a sample of 200 such drivers (n = 200, p = 0.52):

  • n × p = 200 × 0.52 = 104 (≥ 5)

  • q = 1 - 0.52 = 0.48

  • n × q = 200 × 0.48 = 96 (≥ 5)

  • Mean: $\mu = 200 \times 0.52 = 104$

  • Standard deviation: $\sigma = \sqrt{200 \times 0.52 \times 0.48} \approx 7.07$

  • Find the probability that at most 90 drivers admit to aggressive driving (x ≤ 90):

    • Apply continuity correction: x < 90.5

    • Calculate z-score:

    • $z = \frac{90.5 - 104}{7.0654} \approx -1.91$

    • Find P(z < -1.91) using the standard normal table: 0.0281

    • Interpretation: There is approximately a 2.81% chance that at most 90 drivers will admit to aggressive driving.

Summary Table: Steps for Normal Approximation to the Binomial

Step	Description
1	Verify binomial setting (fixed n, independent, constant p)
2	Check n × p ≥ 5 and n × q ≥ 5
3	Calculate mean (μ) and standard deviation (σ)
4	Apply continuity correction to x-values
5	Convert corrected x to z-score: $z = \frac{x_{corr} - \mu}{\sigma}$
6	Find probability using standard normal table or technology

Additional info: The continuity correction is essential for accurate approximation because the binomial is discrete and the normal is continuous. Always check the conditions before applying the normal approximation.