To determine whether the given table shows a probability distribution and if so, find its mean and standard deviation, let's go through the criteria for a probability distribution step-by-step.
1. Criteria for a Probability Distribution:
- The sum of all probabilities must be equal to 1.
- Each individual probability must be between 0 and 1 inclusive.
Let's check each criterion:
- Sum of Probabilities:
- We need to add the given probabilities:
[tex]\( P(\text{Left}) = 0.63 \)[/tex]
[tex]\( P(\text{Right}) = 0.30 \)[/tex]
[tex]\( P(\text{No preference}) = 0.05 \)[/tex]
- Adding these together: [tex]\( 0.63 + 0.30 + 0.05 = 0.98 \)[/tex]
- Check the Range:
- Each probability (0.63, 0.30, 0.05) is between 0 and 1 inclusive.
2. Confirmation:
- The range criterion is satisfied as all probabilities lie between 0 and 1 inclusive.
- However, the sum of the probabilities is [tex]\( 0.98 \)[/tex], which is not equal to 1. Therefore, the table does not show a probability distribution because one of the criteria is not satisfied.
Based on this:
- Does the table show a probability distribution?
- The correct answer is: D. No, the sum of all the probabilities is not equal to 1.
3. Mean and Standard Deviation:
- Since the table does not represent a probability distribution, we do not proceed with calculating the mean and standard deviation as it is irrelevant here.
Therefore:
- We have determined that the table does not show a probability distribution due to the sum of the probabilities being [tex]\( 0.98 \)[/tex], which is not equal to 1.
