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.

c. Is a result of 7 peas with green pods a result that is significantly low? Why or why not?

검증된 단계별 안내

Step 1: Identify the type of probability distribution involved. Since the problem involves a fixed number of trials (16 peas), a constant probability of success (0.75 for green pods), and independent trials, this is a binomial distribution. The random variable X represents the number of peas with green pods.

Step 2: Write the formula for the binomial probability distribution. The probability of exactly k successes in n trials is given by:

P (X = k) = \frac{n!}{k! (n - k)!} \cdot p^{k} \cdot (^{1 - p) n - k}

, where n = 16, k = 7, and p = 0.75.

Step 3: To determine if 7 peas with green pods is significantly low, calculate the mean (μ) and standard deviation (σ) of the binomial distribution. The formulas are:

μ = n \cdot p

and

σ = \sqrt{n \cdot p \cdot (1 - p)}

. Substitute n = 16 and p = 0.75 into these formulas.

Step 4: Use the range rule of thumb to determine if 7 is significantly low. A value is considered significantly low if it is below

μ - 2 σ

. Calculate this threshold using the mean and standard deviation from Step 3.

Step 5: Compare the observed value (7 peas with green pods) to the threshold calculated in Step 4. If 7 is below the threshold, it is significantly low; otherwise, it is not. Provide reasoning based on the comparison.

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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 case, the probability of a pea having green pods is 0.75, and we are looking at 16 trials (offspring). This distribution helps determine the likelihood of observing a specific number of successes, such as 7 green pods.

Significance Level

The significance level, often denoted as alpha (α), is a threshold used to determine whether a result is statistically significant. Commonly set at 0.05, it indicates the probability of rejecting the null hypothesis when it is true. In this context, we would compare the observed result of 7 green pods against the expected distribution to see if it falls below this threshold.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population based on sample data. It involves formulating a null hypothesis (e.g., the number of green pods follows the expected distribution) and an alternative hypothesis. By calculating the probability of observing the data under the null hypothesis, we can determine if the result of 7 green pods is significantly low or not.