https://file.aiquickdraw.com/mathbot/file/172948712797421nbz448.jpg

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### Part (a): Find the mean and standard deviation
We are given:
- Probability of a pea having green pods, \( p = 0.25 \)
- The group size, \( n = 28 \)

#### Mean:
The mean (\( \mu \)) for a binomial distribution is given by:
\[
\mu = n \times p
\]
Substitute the values:
\[
\mu = 28 \times 0.25 = 7 \text{ peas}
\]

#### Standard Deviation:
The standard deviation (\( \sigma \)) for a binomial distribution is given by:
\[
\sigma = \sqrt{n \times p \times (1 - p)}
\]
Substitute the values:
\[
\sigma = \sqrt{28 \times 0.25 \times (1 - 0.25)} = \sqrt{28 \times 0.25 \times 0.75} = \sqrt{5.25} \approx 2.3 \text{ peas}
\]

So:
- Mean \( \mu = 7 \text{ peas} \)
- Standard deviation \( \sigma = 2.3 \text{ peas} \)

### Part (b): Range rule of thumb to find significantly low values
The range rule of thumb states that values significantly low are below \( \mu - 2\sigma \).

Calculate:
\[
\mu - 2\sigma = 7 - 2(2.3) = 7 - 4.6 = 2.4
\]

So, values of 2.4 peas or fewer are considered significantly low.

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Would you like more details or have any questions on this solution?

Here are some related questions for deeper understanding:

1. How is the binomial distribution different from the normal distribution?
2. Can we use the normal approximation for this binomial problem?
3. What would be considered significantly high values in this problem?
4. How does the standard deviation help in interpreting the results?
5. What other real-world problems can be modeled using a binomial distribution?

**Tip**: The range rule of thumb is a quick way to estimate significantly low or high values, but using exact probability methods like the binomial probability formula can give more precise results.

Assume that hybridization experiments are conducted with peas having the property that for offspring, there is a 0.25 probability that a pea has green pods. Assume that the offspring peas are randomly selected in groups of 28. Find the mean and standard deviation, and use the range rule of thumb to find significantly low values.

This problem involves calculating the mean and standard deviation for a binomial distribution where each pea has a 0.25 probability of having green pods. Using a sample size of 28, the solution also applies the range rule of thumb to identify significantly low values.

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