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 24. 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
(b) Use the range rule of thumb to find the values separating results that are significantly low or significantly high.
(c) Is a result of 2 peas with green pods a result that is significantly low? Why or why not?

The hybridization experiment has two possible outcomes, that the peas have green pods or that the peas do not have green pods. The researcher is interested in the peas that have green pods so the study variable is:

X: Number of peas with green pods, in a sample of 24.

If you check the binomial criteria:

1. The number of observations of the trial is fixed. n= 24

2. Each observation in the trial is independent, this means that none of the trials will affect the probability of the next trial.

3. The probability of success in the same from one trial to another. "Having green pods" if the sucess of this experiment and the corresponding probability is p= 0.75

So X≈ Bi (n;ρ)

a.

Under the binomial distribution the mean and standard deviation are:

E(X)= n*p= 24*0.75=18

V(X)= n*p*(1-p)= 24*0.75*0.25=4.5

√V(X)= 2.12

b.

The range rule thumb is a method to estimate the range from the standard deviation. It shows that the range is within 4 standard deviations.

So the range of the variable is 4*√V(X)= 4*2.12= 8.485≅8.49

Range= Xmax-Xmin

If we consider the mean to be exactly in the center of the variable, then the max and min values are:

Max value: E(X) + R/2= 18 + (8.49/2)= 22.245

Min value: E(X) - R/2= 18 - (8.49/2)= 13.755

c.

To see how far of the expected value the observation is, you have to do the following calculation:

(X-E(X))/√V(X)= (2 - 18)/2.12= -7.547

X=2 is 7.547 standard deviations away from the mean, this means that it is an extremly rare value.

I hope it helps!