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 26. 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 26.
the value of the mean is μ = 19.5 peas. (type an integer or a decimal. do not round.)
the value of the standard deviation is σ = 2.2 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 15.1 peas or fewer are significantly low. (round to one decimal place as needed.)
values of 23.9 peas or greater are significantly high. (round to one decimal place as needed.)
c. is a result of 13 peas with green pods a result that is significantly low? why or why not?
the result significantly low, because 13 peas with green pods is peas. (round to one decimal place as needed.)

Explanation:

Step1: Recall binomial - mean formula

For a binomial distribution \(X\sim B(n,p)\), the mean \(\mu = np\). Here, \(n = 26\) (number of trials/sample size) and \(p=0.75\) (probability of success). So, \(\mu=np = 26\times0.75=19.5\).

Step2: Recall binomial - standard - deviation formula

The standard deviation \(\sigma=\sqrt{np(1 - p)}\). Substitute \(n = 26\) and \(p = 0.75\) into the formula. First, \(1-p=1 - 0.75 = 0.25\). Then \(\sigma=\sqrt{26\times0.75\times0.25}=\sqrt{4.875}\approx2.2\).

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

The range - rule - of - thumb for significant low values is \(\mu - 2\sigma\) and for significant high values is \(\mu+2\sigma\).

• Significantly low: \(\mu - 2\sigma=19.5-2\times2.2=19.5 - 4.4 = 15.1\).
• Significantly high: \(\mu+2\sigma=19.5 + 2\times2.2=19.5+4.4 = 23.9\).

Step4: Determine if 13 is significantly low

Since \(13<15.1\), the result of 13 peas with green pods is significantly low.

Answer:

The result is significantly low, because 13 peas with green pods is less than 15.1 peas.