To determine the mean and standard deviation for the number of peas with green pods in groups of 34, we can use the properties of the binomial distribution. In a binomial distribution, the mean ([tex]\(\mu\)[/tex]) and standard deviation ([tex]\(\sigma\)[/tex]) are calculated as follows:
1. Mean ([tex]\(\mu\)[/tex]):
The mean for a binomial distribution is calculated using the formula:
[tex]\[
\mu = n \cdot p
\][/tex]
where:
- [tex]\( n \)[/tex] is the number of trials (in this case, the group size of peas).
- [tex]\( p \)[/tex] is the probability of success on an individual trial (in this case, the probability a pea has green pods).
Given values:
[tex]\[
n = 34
\][/tex]
[tex]\[
p = 0.75
\][/tex]
So, the mean is:
[tex]\[
\mu = 34 \cdot 0.75 = 25.5
\][/tex]
Therefore, the value of the mean [tex]\(\mu\)[/tex] is:
[tex]\[
\mu = 25.5
\][/tex]
2. Standard Deviation ([tex]\(\sigma\)[/tex]):
The standard deviation for a binomial distribution is calculated using the formula:
[tex]\[
\sigma = \sqrt{n \cdot p \cdot (1-p)}
\][/tex]
where:
- [tex]\( n \)[/tex] is the number of trials.
- [tex]\( p \)[/tex] is the probability of success.
- [tex]\( 1-p \)[/tex] is the probability of failure.
Using the same given values:
[tex]\[
n = 34
\][/tex]
[tex]\[
p = 0.75
\][/tex]
[tex]\[
1 - p = 0.25
\][/tex]
So, the standard deviation is:
[tex]\[
\sigma = \sqrt{34 \cdot 0.75 \cdot 0.25} \approx 2.5248762345905194
\][/tex]
Therefore, the value of the mean [tex]\(\mu\)[/tex] for the number of peas with green pods in the groups of 34 is:
[tex]\[ \mu = 25.5 \][/tex]
