Let's solve the problem step by step based on the image.

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.

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.