Answer :
Sure, let's go through the steps to solve the given problem.
### Part (a): Find the Mean and Standard Deviation
The problem states there is a probability of 0.75 that a pea has green pods and that groups of 38 peas are selected.
- Mean ([tex]\(\mu\)[/tex]):
[tex]\[ \mu = n \cdot p \][/tex]
where [tex]\( n \)[/tex] is the number of trials (peas) and [tex]\( p \)[/tex] is the probability of success (green pods).
Given:
[tex]\[ n = 38, \quad p = 0.75 \][/tex]
So the mean is:
[tex]\[ \mu = 38 \cdot 0.75 = 28.5 \][/tex]
- Standard Deviation ([tex]\(\sigma\)[/tex]):
[tex]\[ \sigma = \sqrt{n \cdot p \cdot (1 - p)} \][/tex]
Given:
[tex]\[ n = 38, \quad p = 0.75 \][/tex]
Therefore:
[tex]\[ \sigma = \sqrt{38 \cdot 0.75 \cdot (1 - 0.75)} = \sqrt{38 \cdot 0.75 \cdot 0.25} \approx 2.7 \][/tex]
So, the value of the mean is [tex]\(\mu = 28.5\)[/tex] peas, and the standard deviation is [tex]\(\sigma = 2.7\)[/tex] peas.
### Part (b): Using the Range Rule of Thumb to Determine Significantly Low or High Values
To use the range rule of thumb, we determine significantly low and high values as follows:
- Significantly low values are more than 2 standard deviations below the mean.
- Significantly high values are more than 2 standard deviations above the mean.
#### Significantly Low Values
[tex]\[ \text{Significantly low} = \mu - 2\sigma \][/tex]
Given:
[tex]\[ \mu = 28.5, \quad \sigma = 2.7 \][/tex]
So:
[tex]\[ \text{Significantly low} = 28.5 - 2 \cdot 2.7 = 28.5 - 5.4 = 23.1 \][/tex]
#### Significantly High Values
[tex]\[ \text{Significantly high} = \mu + 2\sigma \][/tex]
Given:
[tex]\[ \mu = 28.5, \quad \sigma = 2.7 \][/tex]
So:
[tex]\[ \text{Significantly high} = 28.5 + 2 \cdot 2.7 = 28.5 + 5.4 = 33.9 \][/tex]
### Conclusion
- Values of [tex]\(23.1\)[/tex] peas or fewer are significantly low.
- Values of [tex]\(33.9\)[/tex] peas or more are significantly high.
Therefore, when using the range rule of thumb:
Values of [tex]\(23.1\)[/tex] peas or fewer are significantly low.
### Part (a): Find the Mean and Standard Deviation
The problem states there is a probability of 0.75 that a pea has green pods and that groups of 38 peas are selected.
- Mean ([tex]\(\mu\)[/tex]):
[tex]\[ \mu = n \cdot p \][/tex]
where [tex]\( n \)[/tex] is the number of trials (peas) and [tex]\( p \)[/tex] is the probability of success (green pods).
Given:
[tex]\[ n = 38, \quad p = 0.75 \][/tex]
So the mean is:
[tex]\[ \mu = 38 \cdot 0.75 = 28.5 \][/tex]
- Standard Deviation ([tex]\(\sigma\)[/tex]):
[tex]\[ \sigma = \sqrt{n \cdot p \cdot (1 - p)} \][/tex]
Given:
[tex]\[ n = 38, \quad p = 0.75 \][/tex]
Therefore:
[tex]\[ \sigma = \sqrt{38 \cdot 0.75 \cdot (1 - 0.75)} = \sqrt{38 \cdot 0.75 \cdot 0.25} \approx 2.7 \][/tex]
So, the value of the mean is [tex]\(\mu = 28.5\)[/tex] peas, and the standard deviation is [tex]\(\sigma = 2.7\)[/tex] peas.
### Part (b): Using the Range Rule of Thumb to Determine Significantly Low or High Values
To use the range rule of thumb, we determine significantly low and high values as follows:
- Significantly low values are more than 2 standard deviations below the mean.
- Significantly high values are more than 2 standard deviations above the mean.
#### Significantly Low Values
[tex]\[ \text{Significantly low} = \mu - 2\sigma \][/tex]
Given:
[tex]\[ \mu = 28.5, \quad \sigma = 2.7 \][/tex]
So:
[tex]\[ \text{Significantly low} = 28.5 - 2 \cdot 2.7 = 28.5 - 5.4 = 23.1 \][/tex]
#### Significantly High Values
[tex]\[ \text{Significantly high} = \mu + 2\sigma \][/tex]
Given:
[tex]\[ \mu = 28.5, \quad \sigma = 2.7 \][/tex]
So:
[tex]\[ \text{Significantly high} = 28.5 + 2 \cdot 2.7 = 28.5 + 5.4 = 33.9 \][/tex]
### Conclusion
- Values of [tex]\(23.1\)[/tex] peas or fewer are significantly low.
- Values of [tex]\(33.9\)[/tex] peas or more are significantly high.
Therefore, when using the range rule of thumb:
Values of [tex]\(23.1\)[/tex] peas or fewer are significantly low.
