Normal Approximations to Binomial Distributions: Study Notes

Normal Probability Distributions

Normal Approximations to Binomial Distributions

This section explores how the normal distribution can be used to approximate binomial probabilities, a key concept in inferential statistics. The approximation is particularly useful when dealing with large sample sizes, where calculating exact binomial probabilities becomes cumbersome.

• 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, bell-shaped probability distribution characterized by its mean and standard deviation.

• Normal Approximation: When certain conditions are met, the binomial distribution can be approximated by a normal distribution, simplifying probability calculations.

Conditions for Normal Approximation

To use the normal distribution as an approximation for the binomial distribution, specific criteria must be satisfied. These ensure the binomial distribution is sufficiently symmetric and bell-shaped.

• Criteria: The approximation is valid when both $np \geq 5$ and $nq \geq 5$, where:

  • $n$ = number of trials

  • $p$ = probability of success

  • $q = 1 - p$ = probability of failure

• Mean: $\mu = np$

• Standard Deviation: $\sigma = \sqrt{npq}$

Example: For $n = 45$, $p = 0.69$, $q = 0.31$, both $np$ and $nq$ are greater than 5, so the normal approximation is valid.

Visualizing the Approximation

As the number of trials ($n$) increases, the binomial distribution's histogram becomes more bell-shaped and closely resembles the normal curve.

• For small $n$ (e.g., $n = 4$), the binomial distribution is discrete and may be skewed.

• For larger $n$ (e.g., $n = 25$ or $n = 50$), the distribution approaches normality.

Continuity Correction

Since the binomial distribution is discrete and the normal distribution is continuous, a continuity correction is applied when approximating binomial probabilities using the normal distribution. This involves adjusting the interval by 0.5 units to include all possible values.

• Correction: Move 0.5 unit to the left and right of the midpoint.

• Example: To approximate the probability of getting between 270 and 310 successes (inclusive), use $269.5 < x < 310.5$.

• For "at least" probabilities: $x > c$ becomes $x > c - 0.5$.

• For "fewer than" probabilities: $x < c$ becomes $x < c - 0.5$.

Steps for Using Normal Approximation

Follow these steps to approximate binomial probabilities using the normal distribution:

1. Check if $np \geq 5$ and $nq \geq 5$.

2. Calculate the mean ($\mu = np$) and standard deviation ($\sigma = \sqrt{npq}$).

3. Apply the continuity correction to the desired interval.

4. Convert the value(s) to z-scores using $z = \frac{x - \mu}{\sigma}$.

5. Use the standard normal table to find the probability.

Examples of Normal Approximation

Several examples illustrate the application of normal approximation to binomial probabilities:

• Example 1: Probability that fewer than 33 out of 45 students graduate ($p = 0.69$). - Apply continuity correction: $x < 32.5$ - Calculate z-score and use standard normal table. - Result: Probability ≈ 0.6808 (68.08%)

• Example 2: Probability that at least 125 out of 200 adults drive weekly ($p = 0.65$). - Apply continuity correction: $x > 124.5$ - Calculate z-score and use standard normal table. - Result: Probability ≈ 0.7939 (79.4%)

• Example 3: Probability that exactly 40 out of 100 NFL players believe they have CTE ($p = 0.344$). - Apply continuity correction: $39.5 < x < 40.5$ - Calculate z-scores for both bounds. - Result: Probability ≈ 0.042 (4.2%)

Summary Table: Binomial vs. Normal Approximation

The following table summarizes the key differences and conditions for using the normal approximation:

Feature	Binomial Distribution	Normal Approximation
Type	Discrete	Continuous
Parameters	$n$, $p$	$\mu = np$, $\sigma = \sqrt{npq}$
Conditions	Any $n$	$np \geq 5$, $nq \geq 5$
Continuity Correction	Not needed	Required

Key Formulas

• Mean: $\mu = np$

• Standard Deviation: $\sigma = \sqrt{npq}$

• Z-score: $z = \frac{x - \mu}{\sigma}$

Applications and Importance

The normal approximation to the binomial distribution is widely used in statistics for hypothesis testing, confidence intervals, and practical probability calculations, especially when sample sizes are large.

• Reduces computational complexity

• Allows use of standard normal tables

• Supports inferential statistics methods

Additional info: The examples and explanations have been expanded for clarity and completeness, ensuring the notes are self-contained and suitable for exam preparation.