Question 1180440: 1.Suppose a population hold the ages of five family members,as follows: 1,13,32,34, and 59.Calculate the population mean and the sample means of the random variable of size n=2. Illustrate the probability distributions of the sample means of the ages of the five family members.
2.Referring to the situation above, construct a sampling distribution of the sample variances and sample standard deviations.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! **1. Population Mean and Sample Means:**
* **Population Mean:**
The population mean (μ) is the average of all the ages:
μ = (1 + 13 + 32 + 34 + 59) / 5 = 139 / 5 = 27.8
* **Sample Means:**
Since the sample size is 2, we need to find all possible combinations of 2 ages from the population. There are 10 such combinations:
(1, 13), (1, 32), (1, 34), (1, 59), (13, 32), (13, 34), (13, 59), (32, 34), (32, 59), (34, 59)
Now, calculate the mean of each sample:
7, 16.5, 17.5, 30, 22.5, 23.5, 36, 33, 45.5, 46.5
* **Probability Distribution of Sample Means:**
Each of the 10 sample means has an equal probability of occurring (1/10). The probability distribution is:
| Sample Mean | Probability |
|---|---|
| 7 | 1/10 |
| 16.5 | 1/10 |
| 17.5 | 1/10 |
| 30 | 1/10 |
| 22.5 | 1/10 |
| 23.5 | 1/10 |
| 36 | 1/10 |
| 33 | 1/10 |
| 45.5 | 1/10 |
| 46.5 | 1/10 |
**2. Sampling Distribution of Sample Variances and Standard Deviations:**
* **Sample Variances:**
For each sample, calculate the variance using the formula:
s² = Σ(xᵢ - x̄)² / (n - 1)
where xᵢ are the ages in the sample, x̄ is the sample mean, and n is the sample size (2).
The sample variances are:
72, 480.5, 544.5, 1682, 180.5, 220.5, 1058, 2, 364.5, 312.5
* **Sample Standard Deviations:**
For each sample, take the square root of the variance:
s = √s²
The sample standard deviations are:
8.49, 21.92, 23.33, 41.01, 13.44, 14.85, 32.53, 1.41, 19.09, 17.68
* **Sampling Distribution:**
Each sample variance and standard deviation has a probability of 1/10 of occurring. The sampling distributions are:
**Sample Variances:**
| Sample Variance | Probability |
|---|---|
| 72 | 1/10 |
| 480.5 | 1/10 |
| 544.5 | 1/10 |
| 1682 | 1/10 |
| 180.5 | 1/10 |
| 220.5 | 1/10 |
| 1058 | 1/10 |
| 2 | 1/10 |
| 364.5 | 1/10 |
| 312.5 | 1/10 |
**Sample Standard Deviations:**
| Sample Standard Deviation | Probability |
|---|---|
| 8.49 | 1/10 |
| 21.92 | 1/10 |
| 23.33 | 1/10 |
| 41.01 | 1/10 |
| 13.44 | 1/10 |
| 14.85 | 1/10 |
| 32.53 | 1/10 |
| 1.41 | 1/10 |
| 19.09 | 1/10 |
| 17.68 | 1/10 |
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