Question 1160974: The data below represent the number of days absent per year in a population of three employees of a small company:
1 4 7
a. Assuming that you sample without replacement, select all possible samples of n = 2 and construct the sampling distribution of the mean. Compare the mean of all the samples means and also compute the population mean. Are they equal? What is the property called?
b. Repeat (a) for all the sampling distribution of the mean in (a) for n= 3. Which sampling distribution has less variability? Why?
c. Assuming that you sample with replacement, repeat (a) through (c) and compare the results. Which sampling distribution have the least variability - those in (a) or (b) ? Why?
Answer by CPhill(2285)   (Show Source):
You can put this solution on YOUR website! ### Population Parameters
* **Population Data:** $1, 4, 7$
* **Population Mean ($\mu$):** $\frac{1 + 4 + 7}{3} = \mathbf{4.0}$
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### Part A: Sampling Without Replacement ($n = 2$)
**1. All Possible Samples and Sample Means ($\bar{x}$):**
* Sample $(1, 4) \implies \bar{x} = \frac{1 + 4}{2} = 2.5$
* Sample $(1, 7) \implies \bar{x} = \frac{1 + 7}{2} = 4.0$
* Sample $(4, 7) \implies \bar{x} = \frac{4 + 7}{2} = 5.5$
**2. Sampling Distribution of the Mean:**
| Sample Mean ($\bar{x}$) | Probability $P(\bar{x})$ |
| --- | --- |
| $2.5$ | $1/3$ |
| $4.0$ | $1/3$ |
| $5.5$ | $1/3$ |
**3. Comparison & Property:**
* **Mean of Sample Means ($\mu_{\bar{x}}$):** $\frac{2.5 + 4.0 + 5.5}{3} = \mathbf{4.0}$
* **Population Mean ($\mu$):** $\mathbf{4.0}$
* **Are they equal?** Yes, $\mu_{\bar{x}} = \mu$.
* **Property:** This property is called **unbiasedness** (i.e., the sample mean $\bar{x}$ is an **unbiased estimator** of the population mean $\mu$).
---
### Part B: Sampling Without Replacement ($n = 3$)
**1. All Possible Samples and Sample Means ($\bar{x}$):**
* Only $1$ sample is possible: $(1, 4, 7)$
* Sample mean: $\bar{x} = \frac{1 + 4 + 7}{3} = \mathbf{4.0}$
**2. Sampling Distribution of the Mean:**
| Sample Mean ($\bar{x}$) | Probability $P(\bar{x})$ |
| --- | --- |
| $4.0$ | $1.0$ |
**3. Comparison of Variability:**
* Variance of sample means for $n = 2$: $\sigma_{\bar{x}}^2 = 1.5$
* Variance of sample means for $n = 3$: $\sigma_{\bar{x}}^2 = 0.0$
* **Which has less variability?** The sampling distribution for **$n = 3$** has less variability (zero variability).
* **Why?** When $n = 3$ without replacement, the sample includes the entire population. Every sample is identical to the population itself, resulting in zero standard error.
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### Part C: Sampling With Replacement ($n = 2$ and $n = 3$)
#### **1. With Replacement ($n = 2$)**
* **$3^2 = 9$ possible samples:**
| Sample | Mean ($\bar{x}$) | Sample | Mean ($\bar{x}$) |
| --- | --- | --- | --- |
| $(1, 1)$ | $1.0$ | $(4, 4)$ | $4.0$ |
| $(1, 4)$ | $2.5$ | $(4, 7)$ | $5.5$ |
| $(1, 7)$ | $4.0$ | $(7, 1)$ | $4.0$ |
| $(4, 1)$ | $2.5$ | $(7, 4)$ | $5.5$ |
| | | $(7, 7)$ | $7.0$ |
* **Sampling Distribution ($n = 2$, With Replacement):**
| $\bar{x}$ | $1.0$ | $2.5$ | $4.0$ | $5.5$ | $7.0$ |
| --- | --- | --- | --- | --- | --- |
| **$P(\bar{x})$** | $1/9$ | $2/9$ | $3/9$ | $2/9$ | $1/9$ |
* **Mean of Sample Means:** $\mu_{\bar{x}} = \mathbf{4.0}$
* **Variance ($\sigma_{\bar{x}}^2$):** $\mathbf{3.0}$
---
#### **2. With Replacement ($n = 3$)**
* **$3^3 = 27$ possible samples.**
* **Mean of Sample Means:** $\mu_{\bar{x}} = \mathbf{4.0}$
* **Variance ($\sigma_{\bar{x}}^2$):** $\mathbf{2.0}$
---
#### **3. Summary & Comparison of Variability Across All Cases**
| Sampling Method | Sample Size ($n$) | Mean ($\mu_{\bar{x}}$) | Variance of $\bar{x}$ ($\sigma_{\bar{x}}^2$) |
| --- | --- | --- | --- |
| **Without Replacement** | $n = 2$ | $4.0$ | $1.5$ |
| **Without Replacement** | $n = 3$ | $4.0$ | **$0.0$** *(Least overall)* |
| **With Replacement** | $n = 2$ | $4.0$ | $3.0$ |
| **With Replacement** | $n = 3$ | $4.0$ | $2.0$ |
* **Which sampling distribution has the least variability?**
* Between (a) and (b), **(b) ($n = 3$ without replacement)** has the **absolute least variability** ($\sigma_{\bar{x}}^2 = 0$).
* Across all non-zero sampling distributions ($n=2$ WOR vs. $n=2$ WR vs. $n=3$ WR), **$n = 2$ without replacement** has less variability than with replacement for the same sample size ($1.5 < 3.0$).
* **Why?**
1. **Larger sample sizes ($n$)** reduce variability because larger samples capture more information about the population, averaging out extreme values (Central Limit Theorem).
2. **Sampling without replacement** further reduces variability by applying the *Finite Population Correction (FPC)* factor $\left(\frac{N-n}{N-1}\right)$, which accounts for the fact that sampling without replacement depletes the remaining population options.
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