Ch. 5 - Discrete Probability Distributions

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 5 - Discrete Probability Distributions

Problem 5.2.32a

Chapter 5, Problem 5.2.32a

In Exercises 31 and 32, assume that hybridization experiments are conducted with peas having the property that for offspring, there is a 0.75 probability that a pea has green pods (as in one of Mendel’s famous experiments).

Hybrids Assume that offspring peas are randomly selected in groups of 16.

a. Find the mean and standard deviation for the numbers of peas with green pods in the groups of 16.

Verified step by step guidance

Step 1: Recognize that this is a binomial distribution problem because there are two possible outcomes for each pea: having green pods (success) or not having green pods (failure). The probability of success (green pods) is given as p = 0.75, and the number of trials (n) is 16.

Step 2: Recall the formula for the mean (μ) of a binomial distribution: μ = n × p. Substitute the given values of n = 16 and p = 0.75 into this formula to calculate the mean.

Step 3: Recall the formula for the standard deviation (σ) of a binomial distribution: σ = √(n × p × (1 - p)). Substitute the given values of n = 16, p = 0.75, and (1 - p) = 0.25 into this formula to calculate the standard deviation.

Step 4: Simplify the expressions for both the mean and the standard deviation. For the mean, multiply n and p. For the standard deviation, calculate the product n × p × (1 - p) and then take the square root of the result.

Step 5: Interpret the results. The mean represents the expected number of peas with green pods in a group of 16, and the standard deviation measures the variability in the number of peas with green pods across different groups of 16.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 case, the success is defined as a pea having green pods, with a probability of 0.75. The distribution is characterized by two parameters: the number of trials (n) and the probability of success (p).

Mean of a Binomial Distribution

The mean of a binomial distribution can be calculated using the formula μ = n * p, where n is the number of trials and p is the probability of success. For this problem, with 16 peas and a probability of 0.75 for green pods, the mean number of peas with green pods can be determined by substituting these values into the formula.

Standard Deviation of a Binomial Distribution

The standard deviation of a binomial distribution is calculated using the formula σ = √(n * p * (1 - p)). This formula accounts for the variability in the number of successes across the trials. In this scenario, it helps quantify how much the number of peas with green pods is expected to deviate from the mean in groups of 16.

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Mean & Standard Deviation of Binomial Distribution

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