Significance with Range Rule of Thumb. In Exercises 31 and 32, 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 (as in one of Mendel’s famous experiments).

Hybrids Assume that offspring peas are randomly selected in groups of 16.

a. Find the mean and standard deviation for the numbers of peas with green pods in the groups of 16.

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 7 peas with green pods a result that is significantly low? Why or why not?

It is given that a group of 16 peas is examined. The probability of a pea with green pods is equal to 0.75.

a.

Here, the probability of a pea with a green pod is given to be equal to p=0.75.

The probability of a pea with any other colored pod is computed below:

$$\begin{gathered} q=1-p \\ =1-0.75 \\ =0.25 \end{gathered}$$

The number of trials (n) is equal to 16.

Thus, the mean value is given as follows:

$$\begin{gathered} \mu =np \\ =\left(16\right)\left(0.75\right) \\ =12.0 \end{gathered}$$

Therefore, the mean number of peas with green pods is equal to 12.0.

The standard deviation is computed below:

$$\begin{gathered} \sigma =\sqrt{npq} \\ =\sqrt{\left(16\right)\left(0.75\right)\left(0.25\right)} \\ =1.7 \end{gathered}$$

Therefore, the standard deviation of the number of peas with green pods is equal to 1.7.

b.

By using the range rule of thumb, the significantly low number of peas with green pods is computed below:

$$\begin{gathered} \mu -2\sigma =12.0-\left(2\right)\left(1.7\right) \\ =8.6 \end{gathered}$$

Thus, the significantly low number of peas with green pods is8.6 or less.

The significantly high number of peas with green pods is computed below:

$$\begin{gathered} \mu +2\sigma =12.0+\left(2\right)\left(1.7\right) \\ =15.4 \end{gathered}$$

Thus, the significantly high number of peas with green pods is15.4 or more.

Moreover, the values that are not significant will lie between 8.6and 15.4.

c.

Here, the value of 7 is less than 8.6.

Therefore, the value of 7 peas with green pods can be considered significantly low as it is less than 8.6.