Certainly! Let's break down the problem step-by-step to determine if this scenario fits a binomial probability distribution and, if so, what the values of [tex]\( n \)[/tex], [tex]\( p \)[/tex], and [tex]\( q \)[/tex] should be.
### Binomial Probability Distribution:
A binomial probability distribution is appropriate if the following conditions are met:
1. The experiment consists of a fixed number of trials, [tex]\( n \)[/tex].
2. Each trial has only two outcomes: success or failure.
3. The probability of success, [tex]\( p \)[/tex], is the same for each trial.
4. The trials are independent.
In this case:
1. The experiment is selecting ten Texans at random, so the number of trials [tex]\( n = 10 \)[/tex].
2. We are interested in whether each selected person is male (success) or not (failure), satisfying the two outcomes condition.
3. According to the Census, [tex]\( 49.65\% \)[/tex] of Texas residents were male, thus the probability of selecting a male, [tex]\( p \)[/tex], is [tex]\( 0.4965 \)[/tex].
4. Each selection is independent of the others.
Therefore, this experiment fits the criteria for a binomial probability distribution.
### Parameters of the Binomial Distribution:
- [tex]\( n \)[/tex]: the number of trials.
- [tex]\( p \)[/tex]: the probability of success (selecting a male).
- [tex]\( q \)[/tex]: the probability of failure, which is [tex]\( 1 - p \)[/tex].
Given these:
- [tex]\( n = 10 \)[/tex]
- [tex]\( p = 0.4965 \)[/tex]
- [tex]\( q = 1 - 0.4965 = 0.5035 \)[/tex]
### Conclusion:
The correct parameters for this binomial distribution, given the problem statement, are:
- [tex]\( n = 10 \)[/tex]
- [tex]\( p = 0.4965 \)[/tex]
- [tex]\( q = 0.5035 \)[/tex]
Therefore, the correct answer is:
1. Yes; [tex]\( n = 10 \)[/tex], [tex]\( p = 0.4965 \)[/tex], [tex]\( q = 0.5035 \)[/tex]
Hence, the best answer from the given choices is D.
