Let's go through each case to determine the mean, variance, and standard deviation for the given values of [tex]\( n \)[/tex] and [tex]\( p \)[/tex] in a binomial distribution.
### a. [tex]\( n = 100 \)[/tex], [tex]\( p = 0.75 \)[/tex]
1. Mean ([tex]\(\mu\)[/tex]):
[tex]\[
\mu = n \cdot p = 100 \cdot 0.75 = 75
\][/tex]
2. Variance ([tex]\(\sigma^2\)[/tex]):
[tex]\[
\sigma^2 = n \cdot p \cdot (1 - p) = 100 \cdot 0.75 \cdot 0.25 = 18.75
\][/tex]
3. Standard Deviation ([tex]\(\sigma\)[/tex]):
[tex]\[
\sigma = \sqrt{\sigma^2} = \sqrt{18.75} \approx 4.33
\][/tex]
### b. [tex]\( n = 300 \)[/tex], [tex]\( p = 0.3 \)[/tex]
1. Mean ([tex]\(\mu\)[/tex]):
[tex]\[
\mu = n \cdot p = 300 \cdot 0.3 = 90
\][/tex]
2. Variance ([tex]\(\sigma^2\)[/tex]):
[tex]\[
\sigma^2 = n \cdot p \cdot (1 - p) = 300 \cdot 0.3 \cdot 0.7 = 63
\][/tex]
3. Standard Deviation ([tex]\(\sigma\)[/tex]):
[tex]\[
\sigma = \sqrt{\sigma^2} = \sqrt{63} \approx 7.94
\][/tex]
### c. [tex]\( n = 20 \)[/tex], [tex]\( p = 0.5 \)[/tex]
1. Mean ([tex]\(\mu\)[/tex]):
[tex]\[
\mu = n \cdot p = 20 \cdot 0.5 = 10
\][/tex]
2. Variance ([tex]\(\sigma^2\)[/tex]):
[tex]\[
\sigma^2 = n \cdot p \cdot (1 - p) = 20 \cdot 0.5 \cdot 0.5 = 5
\][/tex]
3. Standard Deviation ([tex]\(\sigma\)[/tex]):
[tex]\[
\sigma = \sqrt{\sigma^2} = \sqrt{5} \approx 2.24
\][/tex]
### d. [tex]\( n = 10 \)[/tex], [tex]\( p = 0.8 \)[/tex]
1. Mean ([tex]\(\mu\)[/tex]):
[tex]\[
\mu = n \cdot p = 10 \cdot 0.8 = 8
\][/tex]
2. Variance ([tex]\(\sigma^2\)[/tex]):
[tex]\[
\sigma^2 = n \cdot p \cdot (1 - p) = 10 \cdot 0.8 \cdot 0.2 = 1.6
\][/tex]
3. Standard Deviation ([tex]\(\sigma\)[/tex]):
[tex]\[
\sigma = \sqrt{\sigma^2} = \sqrt{1.6} \approx 1.26
\][/tex]
### e. [tex]\( n = 1000 \)[/tex], [tex]\( p = 0.1 \)[/tex]
1. Mean ([tex]\(\mu\)[/tex]):
[tex]\[
\mu = n \cdot p = 1000 \cdot 0.1 = 100
\][/tex]
2. Variance ([tex]\(\sigma^2\)[/tex]):
[tex]\[
\sigma^2 = n \cdot p \cdot (1 - p) = 1000 \cdot 0.1 \cdot 0.9 = 90
\][/tex]
3. Standard Deviation ([tex]\(\sigma\)[/tex]):
[tex]\[
\sigma = \sqrt{\sigma^2} = \sqrt{90} \approx 9.49
\][/tex]
### Summary
- a. [tex]\( n = 100, p = 0.75 \rightarrow \mu = 75, \sigma^2 = 18.75, \sigma \approx 4.33 \)[/tex]
- b. [tex]\( n = 300, p = 0.3 \rightarrow \mu = 90, \sigma^2 = 63, \sigma \approx 7.94 \)[/tex]
- c. [tex]\( n = 20, p = 0.5 \rightarrow \mu = 10, \sigma^2 = 5, \sigma \approx 2.24 \)[/tex]
- d. [tex]\( n = 10, p = 0.8 \rightarrow \mu = 8, \sigma^2 = 1.6, \sigma \approx 1.26 \)[/tex]
- e. [tex]\( n = 1000, p = 0.1 \rightarrow \mu = 100, \sigma^2 = 90, \sigma \approx 9.49 \)[/tex]
