To address the question of finding the center and shape of the given binomial distribution with parameters [tex]\( n = 10 \)[/tex] and [tex]\( p = 0.8 \)[/tex], follow these steps:
1. Calculate the Center of the Distribution:
The center of a binomial distribution is given by its mean, which can be calculated using the formula:
[tex]\[ \text{Mean} = n \times p \][/tex]
Given:
- [tex]\( n = 10 \)[/tex]
- [tex]\( p = 0.8 \)[/tex]
Substitute these values into the formula:
[tex]\[ \text{Mean} = 10 \times 0.8 = 8.0 \][/tex]
Therefore, the center of the distribution is 8.
2. Determine the Shape of the Distribution:
To determine the shape of the binomial distribution, examine the product [tex]\( np(1 - p) \)[/tex], often denoted as [tex]\( npq \)[/tex], where [tex]\( q \)[/tex] is [tex]\( 1 - p \)[/tex].
Calculate [tex]\( q \)[/tex]:
[tex]\[ q = 1 - p = 1 - 0.8 = 0.2 \][/tex]
Now, find [tex]\( npq \)[/tex]:
[tex]\[ npq = n \times p \times q = 10 \times 0.8 \times 0.2 = 1.6 \][/tex]
For binomial distributions:
- If [tex]\( npq > 10 \)[/tex], the distribution is approximately Normal.
- If [tex]\( npq \leq 10 \)[/tex], the distribution can be skewed.
In this case, [tex]\( npq = 1.6 \)[/tex], which is less than 10. This indicates the distribution is not approximately Normal. Instead, it is skewed.
Given the values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex], since [tex]\( p = 0.8 \)[/tex] is significantly greater than [tex]\( q = 0.2 \)[/tex], the distribution will be skewed left (more outcomes are concentrated on the higher side of the value range).
Therefore, the shape of the distribution is skewed left.
Conclusion:
The center of the given binomial distribution is 8, and the shape is skewed left. Thus, the correct answer is:
- Center: 8
- Shape: skewed left
Hence, the detailed answer is:
Center: 8
Shape: skewed left
