A previous study found that $80\%$ of people preferred drinking Pepsi over Coca Cola. Use a normal distribution to approximate the probability that, from this same random sample of $100$ people, that between $10$ and $11$ people prefer Coca Cola.

• A. 0.0125
• B. 0.9875
• C. 0.8
• D. 0.105

Answer: (A)

Verified step by step guidance

1. Identify the problem as a binomial distribution problem where the probability of success (preferring Coca Cola) is 0.2 (since 80% prefer Pepsi, 20% prefer Coca Cola).
2. Use the normal approximation to the binomial distribution. For a binomial distribution with parameters n (number of trials) and p (probability of success), the mean (μ) is given by μ = np and the standard deviation (σ) is given by σ = sqrt(np(1-p)).
3. Calculate the mean: μ = 100 * 0.2 = 20.
4. Calculate the standard deviation: σ = sqrt(100 * 0.2 * 0.8) = sqrt(16) = 4.
5. Convert the binomial problem to a normal distribution problem by finding the z-scores for 10 and 11 using the formula z = (X - μ) / σ, where X is the number of successes. Then, use the standard normal distribution to find the probability that the number of people preferring Coca Cola is between 10 and 11.