Question

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 μ = peas (type an integer or a decimal. do not round.)
the value of the standard deviation is σ = 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 peas or fewer are significantly low (round to one decimal place as needed.)
values of peas or greater are significantly high (round to one decimal place as needed.)
c. is a result of 18 peas with green pods a result that is significantly low? why or why not?
the result significantly low because 18 peas with green pods is peas (round to one decimal place as needed.)

Explanation:

Step1: Calculate the mean

For a binomial distribution, the mean $\mu = np$, where $n = 36$ (number of trials) and $p=0.75$ (probability of success). So, $\mu=36\times0.75 = 27$.

Step2: Calculate the standard - deviation

The standard deviation for a binomial distribution is $\sigma=\sqrt{np(1 - p)}$. Substitute $n = 36$ and $p = 0.75$ into the formula. First, $1-p=1 - 0.75=0.25$. Then $\sigma=\sqrt{36\times0.75\times0.25}=\sqrt{6.75}\approx2.6$.

Step3: Use the range - rule of thumb for significant values

The range - rule of thumb states that significant low values are $\mu - 2\sigma$ and lower, and significant high values are $\mu+2\sigma$ and higher.
$\mu - 2\sigma=27-2\times2.6=27 - 5.2 = 21.8$.
$\mu + 2\sigma=27+2\times2.6=27 + 5.2 = 32.2$.

Step4: Determine if 18 is significantly low

Since $18<21.8$, 18 is significantly low.

Answer:

a. The value of the mean is $\mu = 27$ peas.
The value of the standard deviation is $\sigma\approx2.6$ peas.
b. Values of $21.8$ peas or fewer are significantly low.
Values of $32.2$ peas or greater are significantly high.
c. The result is significantly low because 18 peas with green pods is less than $21.8$ peas.