To determine the sample mean and the corresponding probability for the outcome [tex]\( (2, 2, 0) \)[/tex], let's work through the problem step by step.
1. Calculate the Sample Mean:
The sample mean of a set of outcomes is found by summing all the outcomes and then dividing by the number of outcomes. For the outcome [tex]\( (2, 2, 0) \)[/tex]:
[tex]\[
\text{Sample Mean} = \frac{2 + 2 + 0}{3} = \frac{4}{3}
\][/tex]
2. Compare the Sample Mean with Given Values:
We are provided with three sets of sample means and probabilities:
1. [tex]\(\varepsilon = \frac{4}{3}, p = 0.025\)[/tex]
2. [tex]\(\bar{x} = \frac{4}{3}, p = 0.05\)[/tex]
3. [tex]\(8 = 4, p = 0.025\)[/tex]
Among these, we need to find which set of sample mean matches our calculated sample mean.
Our calculated sample mean is [tex]\(\frac{4}{3} = 1.3333333333333333\)[/tex].
3. Identify the Matching Probability:
Comparing our calculated sample mean (1.3333333333333333) to the values in the provided sets, we see:
- [tex]\(\varepsilon = \frac{4}{3}, p = 0.025\)[/tex] matches
- [tex]\(\bar{x} = \frac{4}{3}, p = 0.05\)[/tex] also matches
- [tex]\(8 = 4, p = 0.025\)[/tex] does not make sense as a sample mean.
It appears that both [tex]\(\varepsilon = \frac{4}{3}\)[/tex] and [tex]\(\bar{x} = \frac{4}{3}\)[/tex] correspond to our calculated sample mean. Since both provide matching sample means, the lowest probability of [tex]\(0.025\)[/tex] associated with [tex]\(\varepsilon\)[/tex] should be used.
Thus, for the outcome [tex]\((2, 2, 0)\)[/tex]:
- The sample mean is [tex]\(1.3333333333333333\)[/tex].
- The corresponding probability is [tex]\(0.025\)[/tex].
