Suppose
𝑋
∼
𝐵
𝑖
𝑛
𝑜
𝑚
(
𝑛
=
10
,
𝑝
)
and we observe a random sample of size 3:
𝑥
1
=
0
,
𝑥
2
=
3
, and
𝑥
3
=
1
.

The average of the observed values (4/3) is a statistic .

Question

Suppose 
𝑋
∼
𝐵
𝑖
𝑛
𝑜
𝑚
(
𝑛
=
10
,
𝑝
)
 and we observe a random sample of size 3: 
𝑥
1
=
0
,
𝑥
2
=
3
, and 
𝑥
3
=
1
.

The average of the observed values (4/3) is a statistic .

BlackTom AI · Accepted Answer

A statistic is any function of the observed sample values. Here the sample mean is computed from the data: (x1 + x2 + x3)/3 = (0 + 3 + 1)/3.

Step-by-step calculation:
1. Sum the observed values: 0 + 3 + 1 = 4.
2. Divide by the sample size (3): 4 / 3 ≈ 1.333...
Thus the observed average is 4/3.

Relevant concepts:
- Random variable model: each Xi ~ Binomial(n = 10, p). The binomial distribution has population (true) mean E[Xi] = 10p, which depends on the unknown parameter p.
- Statistic: any quantity computed from sample data alone (e.g., sample mean, sample variance). A statistic is a function of observed Xi’s and does not involve unknown population parameters.
- Parameter: a fixed but unknown quantity describing the population (here p or 10p). Parameters are not computed from the observed sample.

Why this is a statistic (and not a parameter): The value 4/3 comes directly from the observed sample values and does not require knowledge of p. It is the outcome of the statistic S = (X1 + X2 + X3)/3 evaluated at the observed data. Although the sample mean can serve as an estimator for the population mean (10p), the observed 4/3 itself is a statistic. Alternative labels such as “parameter” are incorrect because a parameter is an intrinsic population quantity (like p or 10p) and is not computed from sample observations.

Conclusion: The average 4/3 is a statistic. This confirms the provided answer: statistic.

Suppose 𝑋 ∼ 𝐵 𝑖 𝑛 𝑜 𝑚 ( 𝑛 = 10 , 𝑝 ) and we observe a random sample of size 3: 𝑥 1 = 0 , 𝑥 2 = 3 , and 𝑥 3 = 1 . The average of the observed values (4/3) is a statistic .多重下拉选择题

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