Using Normal Distribution to Estimate Binomial Probabilities

Solution & Explanation

The normal distribution can be used as an approximation for calculating binomial probabilities under certain conditions, primarily when the sample size is large enough. This approach leverages the Central Limit Theorem (CLT), which states that the distribution of sample means approaches a normal distribution as the sample size increases. Here is a detailed breakdown of the process:

Conditions for Approximation

1. Sample Size & Success Probability:
   • To use a normal distribution to approximate a binomial distribution, it is essential to ensure that the sample size $n$ is sufficiently large.
   • The rule of thumb is that both $n \times p$ and $n \times (1 - p)$ should be greater than 5.
   • Here, $n$ is the number of trials, and $p$ is the probability of success in each trial.

Calculating Mean and Standard Deviation

2. Mean and Standard Deviation:
   • Once the conditions are met, calculate the mean ($\mu$) and the standard deviation ($\sigma$) of the binomial distribution.
   • The mean is calculated as: $\mu = n \times p$
   • The variance is: $\sigma^2 = n \times p \times (1 - p)$
   • The standard deviation is the square root of the variance: $\sigma = \sqrt{n \times p \times (1 - p)}$

Applying Continuity Correction

3. Continuity Correction:
   • Since the normal distribution is continuous and the binomial distribution is discrete, a continuity correction is applied to improve the approximation.
   • This involves adjusting the discrete binomial variable by $\pm 0.5$ depending on the probability being calculated:
     • For $P(X = x)$, use $P(x - 0.5 < X < x + 0.5)$.
     • For $P(X > x)$, use $P(X > x + 0.5)$.
     • For $P(X \leq x)$, use $P(X < x + 0.5)$.
     • For $P(X < x)$, use $P(X < x - 0.5)$.
     • For $P(X \geq x)$, use $P(X > x - 0.5)$.

Calculating the Z-Score

4. Z-Score Calculation:
   • Convert the binomial variable to a standard normal variable (z-score) using the formula: $z = \frac{x - \mu}{\sigma}$
   • Here, $x$ is the value of the binomial variable adjusted by the continuity correction.

Using the Z-Table

5. Lookup Z-Score:
   • Finally, use the z-score to find the corresponding probability from the standard normal distribution table (z-table).
   • This probability provides an approximation of the binomial probability.

By following these steps, the normal distribution can be effectively used to approximate binomial probabilities, especially when the sample size is large, and the success probability is moderate to large. This method is particularly useful in simplifying calculations and providing insights into binomial processes.