Section 7.4: The Normal Approximation to the Binomial Probability Distribution

Section 7.4: The Normal Approximation to the Binomial Probability Distribution

Objective: Approximate Binomial Probabilities Using the Normal Distribution

This section explores how the normal distribution can be used to approximate probabilities for binomial random variables under certain conditions. This is a useful technique when the binomial distribution is difficult to compute directly, especially for large sample sizes.

Criteria for a Binomial Probability Experiment

• Fixed Number of Trials (n): The experiment consists of a fixed number of independent trials.

• Two Possible Outcomes: Each trial results in one of two outcomes, commonly labeled "success" and "failure."

• Constant Probability (p): The probability of success remains the same for each trial.

Conditions for Normal Approximation

• Large Sample Size: The binomial random variable can be approximated by a normal distribution when both $np \geq 10$ and $n(1-p) \geq 10$.

• Independence: Trials must be independent.

When these conditions are met, the binomial distribution is approximately normal, allowing for easier probability calculations.

Mean and Standard Deviation of the Binomial Distribution

• Mean ($\mu$): $\mu = np$

• Standard Deviation ($\sigma$): $\sigma = \sqrt{np(1-p)}$

These formulas are used to match the binomial distribution to a normal model for approximation.

Continuity Correction

When using the normal distribution to approximate binomial probabilities, a continuity correction is applied to account for the fact that the binomial is discrete and the normal is continuous. This involves adjusting the endpoints by 0.5 units.

• For $P(X \leq k)$: Use $P(X \leq k + 0.5)$

• For $P(X \geq k)$: Use $P(X \geq k - 0.5)$

• For $P(a \leq X \leq b)$: Use $P(a - 0.5 \leq X \leq b + 0.5)$

Example 1: Normal Approximation to a Binomial Random Variable

According to the American Red Cross, 7% of people in the United States have blood type O-negative. What is the probability that in a simple random sample of 500 people in the United States, fewer than 36 have blood type O-negative?

• Step 1: Check conditions: $np = 500 \times 0.07 = 35$, $n(1-p) = 500 \times 0.93 = 465$ (both > 10).

• Step 2: Calculate mean and standard deviation:

  • $\mu = np = 35$

  • $\sigma = \sqrt{np(1-p)} = \sqrt{35 \times 0.93} \approx 5.70$

• Step 3: Apply continuity correction: $P(X < 36) \approx P(X \leq 35.5)$

• Step 4: Convert to z-score: $z = \frac{35.5 - 35}{5.70} \approx 0.09$

• Step 5: Use the standard normal table to find the probability.

Note: The approximation using the normal model is only off by 0.0037 from the exact probability computed using technology. Also, the shape of the distribution in the StatCrunch output should be noted.

Example 2: A Normal Approximation in the Binomial

According to the Gallup Organization, 60% of adult Americans are in favor of the death penalty for individuals convicted of murder. In a random sample of 1000 adults in Will County, Illinois, 630 are in favor of the death penalty for individuals convicted of murder.

• Part A: Assuming that 60% of adult Americans in Will County are in favor of the death penalty, what is the probability of obtaining a random sample of 1000 adults with at least 630 Americans in favor?

• Part B: Does the result from part (A) contradict the Gallup Organization's findings? Explain.

To solve, use the normal approximation with continuity correction for $P(X \geq 630)$, calculate the z-score, and interpret the result in context.

Summary Table: Binomial vs. Normal Approximation

Feature	Binomial Distribution	Normal Approximation
Type	Discrete	Continuous
Parameters	n (trials), p (success probability)	$\mu = np$, $\sigma = \sqrt{np(1-p)}$
Conditions for Use	Any n, p	$np \geq 10$, $n(1-p) \geq 10$
Continuity Correction	Not needed	Required

Additional info: The notes include practice questions and examples to reinforce understanding of the normal approximation to the binomial distribution, including the use of continuity correction and interpretation of results in context.