MAT 3: SAMPLING DISTRIBUTION
Sampling Distribution
The Sampling distribution of sample means is a frequency distribution using the means computed from all possible random samples of a specific size taken from a population. The means of the samples are less than or greater than the mean of the population.
SAMPLING ERROR
- The difference between the sample mean and the population
Steps in Constructing the Sampling Distribution of the Means
Determine the number of sets of all possible random samples that can be drawn from the given population by using the formula, NCn, where N is the population size and n is the sample size.
List all the possible samples and compute the mean of each sample.
Construct the sampling distribution.
Construct a histogram of the sampling distribution of the means.
]]Example. A population consists of the numbers 2, 4, 9, 10, and 5. Let us list all possible sample sizes of 3 from this population and compute the mean of each sample.]]
Determine the number of sets of all possible random samples that can be drawn from the given population by using the formula, NCn, where N is the population size and n is the sample size.
List all the possible samples and compute the mean of each sample.
NCn = 10
2, 4, 9, 10 , 5
| sample | 2, 4, 9 | 2, 4, 10 | 2, 4, 5 | 2, 9, 10 | 2, 9, 5 | 2, 10, 5 | 4, 9, 10 | 4, 9, 5 | 4, 10, 5 | 9, 10, 5 |
|---|---|---|---|---|---|---|---|---|---|---|
| Mean | 5 | 5.33 | 3.67 | 7 | 5.33 | 5.67 | 7.67 | 6 | 6.33 | 8 |
- Construct the sampling distribution. (sample mean should be lowest to highest)
| Sample Mean (x) | Frequency | Probability |
|---|---|---|
| 3.67 | 1 | 1/10 = .10 |
| 5 | 1 | 1/10 = .10 |
| 5.33 | 2 | 2/10 = .20 |
| 5.67 | 1 | 1/10 = .10 |
| 6 | 1 | 1/10 = .10 |
| 6.33 | 1 | 1/10 = .10 |
| 7 | 1 | 1/10 = .10 |
| 7.67 | 1 | 1/10 = .10 |
| 8 | 1 | 1/10 = .10 |
| total | 10 | 1.00 |
Construct a Histogram
]]Example 2. Adrian Cedrick receives 82 or 83 as his grade on his three major subjects. Construct the sampling distribution of his mean grade.]]
List all the possible samples.
| A | B | C | A | B | C | |
|---|---|---|---|---|---|---|
| 82 | 82 | 82 | 83 | 82 | 82 | |
| 82 | 82 | 83 | 83 | 82 | 83 | |
| 82 | 83 | 82 | 83 | 83 | 82 | |
| 82 | 83 | 83 | 83 | 83 | 83 |
- Compute the mean of each samples.
| A | B | C | x (divide by 3(subjects) |
|---|---|---|---|
| 82 | 82 | 82 | 82 |
| 82 | 82 | 83 | 82.33 |
| 82 | 83 | 82 | 82.33 |
| 82 | 83 | 83 | 82.67 |
| 83 | 82 | 82 | 82.33 |
| 83 | 82 | 83 | 82.67 |
| 83 | 83 | 82 | 82.67 |
| 83 | 83 | 83 | 83 |
- Construct the sampling distribution. (sample mean should be lowest to highest)
| Sample Mean | Frequency | P (x) |
|---|---|---|
| 82 | 1 | 1/8 = .125 |
| 82.33 | 3 | 3/8 = .375 |
| 82.67 | 3 | 3/8 = .375 |
| 83 | 1 | 1/8 = .125 |
| TOTAL | 8 | 1.00 |
- What is the probability that his mean grade is lower than 83?
P(x < 83) = .125 + .375 + .375
P (x < 83) = .875 / 87.6%
- What is the probability mean that his grade is higher than 82.33?
P(x>82.33) = 3.75 + .125
P (x>82.33) = .5 . 50%