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. Assume that the offspring peas are randomly selected in groups of 36 . Complete parts (a) through (c) below.
a. Find the mean and the standard deviation for the numbers of peas with green pods in the groups of 36 .
The value of the mean is $\mu=27$ peas.
(Type an integer or a decimal. Do not round.)
The value of the standard deviation is $\sigma=2.60$ peas.
(Round to one decimal place as needed.)
b. Use the range rule of thumb to find the values separating results that are significantly low or significantly high.
Values of 21.8 peas or fewer are significantly low.
(Round to one decimal place as needed.)
Values of 32.2 peas or greater are significantly high.
(Round to one decimal place as needed.)
c. Is a result of 31 peas with green pods a result that is significantly high? Why or why not?
The result siynificantly high because 31 peas with green pods is peas. (Round to one decimal place as needed.)

Step 1 ：Given that the probability of a pea having green pods is 0.75 and the number of peas in each group is 36, we can use the formulas for the mean and standard deviation of a binomial distribution to find these values. The mean is given by \(\mu = np\) and the standard deviation is given by \(\sigma = \sqrt{np(1-p)}\).

Step 2 ：Substituting the given values into these formulas, we find that the mean number of peas with green pods in the groups of 36 is \(\mu = 36 \times 0.75 = 27\) peas and the standard deviation is \(\sigma = \sqrt{36 \times 0.75 \times (1-0.75)} = 2.6\) peas.

Step 3 ：We can use the range rule of thumb to find the values separating results that are significantly low or significantly high. This rule states that most values should lie within 2 standard deviations of the mean. So, values that are significantly low are less than \(\mu - 2\sigma\) and values that are significantly high are greater than \(\mu + 2\sigma\).

Step 4 ：Substituting the calculated values into these formulas, we find that values of \(27 - 2 \times 2.6 = 21.8\) peas or fewer are significantly low and values of \(27 + 2 \times 2.6 = 32.2\) peas or greater are significantly high.

Step 5 ：To determine if a result of 31 peas with green pods is significantly high, we compare it to the value for significantly high results. Since 31 is less than 32.2, it is not considered significantly high.

Step 6 ：Final Answer: The mean number of peas with green pods in the groups of 36 is \(\boxed{27}\) peas. The standard deviation is \(\boxed{2.6}\) peas. Values of \(\boxed{21.8}\) peas or fewer are significantly low. Values of \(\boxed{32.2}\) peas or greater are significantly high. A result of 31 peas with green pods is not significantly high.