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 32. Complete parts​ (a) through​ (c) below.
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Part 1
a. Find the mean and the standard deviation for the numbers of peas with green pods in the groups of 32.
The value of the mean is muequals

enter your response here peas.
​(Type an integer or a decimal. Do not​ round.)

In this case, we can model the number of peas with green pods as a binomial distribution. The binomial distribution has two parameters: the number of trials \( n \) and the probability of success on each trial \( p \).

- \( n = 32 \) (since there are 32 offspring peas in the group),
- \( p = 0.75 \) (the probability that a pea has green pods).

### a) To find the mean (\(\mu\)) of a binomial distribution, use the formula:
\[
\mu = n \cdot p
\]
Substitute the values:
\[
\mu = 32 \cdot 0.75 = 24
\]
So, the mean is \( \mu = 24 \) peas.

Do you need details about how to find the standard deviation, or any other clarification?

Here are some related questions to further explore:
1. How do you calculate the standard deviation for a binomial distribution?
2. What would be the variance for this binomial distribution?
3. How would the mean change if \( n \) were increased to 50?
4. What is the probability of getting exactly 24 peas with green pods in a group of 32?
5. How does the binomial distribution differ from the normal distribution?

**Tip:** For a binomial distribution, the standard deviation helps measure the spread of the outcomes around the mean, and it is closely tied to both the number of trials and the success probability.

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 32. 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 32.

Discover how to calculate the mean and standard deviation for peas with green pods using binomial distribution. This problem explores key concepts in probability and statistics, with a focus on using the formulas for mean and standard deviation. Ideal for students in Grades 10-12.

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