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.


b. Use the range rule of thumb to find the values separating results that are significantly low or significantly high.

Verified step by step guidance

1. Step 1: Understand the problem. The range rule of thumb is used to determine values that are significantly low or significantly high. This involves calculating the mean (μ) and standard deviation (σ) of the binomial distribution, and then using the rule: 'Significantly low values are below μ - 2σ, and significantly high values are above μ + 2σ.'
2. Step 2: Identify the parameters of the binomial distribution. The number of trials (n) is 16, and the probability of success (p) is 0.75. The probability of failure (q) is 1 - p = 0.25.
3. Step 3: Calculate the mean (μ) of the binomial distribution using the formula: μ = n × p. Substitute the values of n and p into the formula.
4. Step 4: Calculate the standard deviation (σ) of the binomial distribution using the formula: σ = √(n × p × q). Substitute the values of n, p, and q into the formula.
5. Step 5: Apply the range rule of thumb. Compute the significantly low threshold as μ - 2σ and the significantly high threshold as μ + 2σ. These values will separate results that are significantly low or significantly high.

Key Concepts

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, the probability of 0.75 indicates that there is a 75% chance that a randomly selected pea will have green pods. Understanding probability is essential for analyzing outcomes in hybridization experiments and determining the expected distribution of traits in offspring.

Range Rule of Thumb

The range rule of thumb is a statistical guideline used to identify significant deviations in data. It suggests that the range of a dataset can be estimated as four times the standard deviation. This rule helps in determining the thresholds for what constitutes significantly low or high results, allowing researchers to assess whether observed outcomes in the pea experiments fall within expected variability.

Binomial Distribution

The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this case, the offspring peas can be modeled using a binomial distribution since each pea can either have green pods (success) or not (failure). Understanding this distribution is crucial for calculating probabilities and applying the range rule of thumb in the context of the given hybridization experiments.