To solve this problem, we'll break down the given data and calculate each part step by step.

Part (a): Mean and Standard Deviation

We are dealing with a binomial distribution since there are two possible outcomes for each pea (green pods or not green pods).

Given:

• Probability of a pea having green pods, $p = 0.75$
• Number of trials (peas in the group), $n = 40$

The mean $\mu$ for a binomial distribution is calculated as:

$\mu = n \times p = 40 \times 0.75 = 30$

So the mean $\mu = 30$, which is already given.

The standard deviation $\sigma$ for a binomial distribution is calculated as:

$\sigma = \sqrt{n \times p \times (1 - p)} = \sqrt{40 \times 0.75 \times (1 - 0.75)} = \sqrt{40 \times 0.75 \times 0.25}$

$\sigma = \sqrt{40 \times 0.1875} = \sqrt{7.5} \approx 2.7$

So, the standard deviation $\sigma = 2.7$, which is given.

Part (b): Range Rule of Thumb

The range rule of thumb estimates the significantly low and significantly high values as being within two standard deviations of the mean.

• Significantly low values: These are values that are below $\mu - 2\sigma$.
• Significantly high values: These are values that are above $\mu + 2\sigma$.

Let’s calculate:

$\text{Low threshold} = \mu - 2\sigma = 30 - 2 \times 2.7 = 30 - 5.4 = 24.6$

Rounding to one decimal place, the significantly low value is 24.6 peas or fewer.

$\text{High threshold} = \mu + 2\sigma = 30 + 2 \times 2.7 = 30 + 5.4 = 35.4$

Rounding to one decimal place, the significantly high value is 35.4 peas or more.

Summary:

• The mean is $30$ peas.
• The standard deviation is $2.7$ peas.
• Values of 24.6 peas or fewer are considered significantly low.
• Values of 35.4 peas or more are considered significantly high.

Would you like any further details or have any questions?

Here are 5 related questions:

1. How does the range rule of thumb help in identifying outliers in a data set?
2. What happens to the standard deviation if the sample size increases to 80 peas?
3. Can you explain how the binomial distribution differs from a normal distribution?
4. What are the implications of a higher probability, say $p = 0.9$, on the mean and standard deviation?
5. How would the results change if the number of peas was reduced to 20?

Tip: The range rule of thumb is a quick method, but for more accurate results, you can use precise statistical methods like z-scores or confidence intervals.