One of Mendel’s famous experiments with peas resulted in 580 offspring, and 152 of them were yellow peas. Mendel claimed that under the same conditions, 25% of offspring peas would be yellow. Assume that Mendel’s claim of 25% is true, and assume that a sample consists of 580 offspring peas.

c. Find the probability of 152 or more yellow peas.

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Step 1: Identify the problem as a binomial probability problem. The number of yellow peas (successes) follows a binomial distribution because there are a fixed number of trials (580 peas), two possible outcomes (yellow or not yellow), and a constant probability of success (25% or 0.25).

Step 2: Define the parameters of the binomial distribution. The number of trials (n) is 580, and the probability of success (p) is 0.25. The random variable X represents the number of yellow peas, and we are interested in finding P(X ≥ 152).

Step 3: Use the normal approximation to the binomial distribution. Since n is large and p is not too close to 0 or 1, the binomial distribution can be approximated by a normal distribution with mean μ = n * p and standard deviation σ = √(n * p * (1 - p)). Calculate μ and σ using the formulas: μ = 580 * 0.25 and σ = √(580 * 0.25 * 0.75).

Step 4: Apply the continuity correction. To approximate P(X ≥ 152) using the normal distribution, adjust the value to include the continuity correction: P(X ≥ 152) ≈ P(Z ≥ (152 - μ + 0.5) / σ), where Z is the standard normal variable.

Step 5: Standardize the value and use the standard normal distribution table. Compute the z-score using the formula Z = (152 - μ + 0.5) / σ. Then, use the standard normal distribution table or a statistical software to find the probability corresponding to the calculated z-score.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, the 'success' is the occurrence of yellow peas, with a probability of 0.25. The distribution is characterized by two parameters: the number of trials (n) and the probability of success (p).

Normal Approximation

For large sample sizes, the binomial distribution can be approximated by a normal distribution due to the Central Limit Theorem. This approximation is valid when both np and n(1-p) are greater than 5. In this case, with 580 trials and a probability of 0.25, the normal approximation can simplify the calculation of probabilities for the number of yellow peas.

Cumulative Probability

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a specific value. To find the probability of observing 152 or more yellow peas, one would calculate the cumulative probability for 151 yellow peas and subtract it from 1. This approach allows us to determine the likelihood of achieving a certain outcome in a binomial setting.