Sampling Distribution of Sample Means
Activity 1. Can You Still Recall? (Do it in your notebook)
Evaluate the following: ( Combination of N objects taken n at a time )
1.
6C2
3. 10C4
2.
20C5
4. 8C3
𝑁!
Use this formula to find the combination: NCn = 𝑛!(𝑁−𝑛)! where N > n
Example:
5C2 =
5!
2! (5 − 2)!
5C2 =
5!
2! (3)!
5C2 =
5∗4∗3!
Cancel 3! in the numerator and denominator
2∗1 (3)!
since 3! divide 3! Is 1
5C2 =
5∗4
2∗1
5C2 =
20
2
5C2 =
10
Sampling Distribution of Sample Means (without replacement)
A sampling distribution of sample means is a probability distribution where all possible random samples
of a specific size were taken from a population.
A finite population is a population that has a fixed number of elements or observations
Sampling Error - the difference between sample mean and the population
Steps in Constructing Sampling Distribution of the Means
1. 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.
2. List all the possible samples and compute the mean of each sample.
3. Construct the sampling distribution.
4. Construct a histogram of the sampling distribution of the means.
Example: A population consists if the number 2,4,9,10 and 5. Let us
list all possible sample size of 3 from this population and
compute the mean of each sample.
1. 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.
N = 5 (number of population)
n=3
NCn =
𝑁!
𝑛! (𝑁 − 𝑛)!
5C3 =
5!
3! (5 − 3)!
5C3 =
5C3 =
5!
3! (2)!
5 ∗ 4 ∗ 3 ∗ 2!
3 ∗ 2 ∗ 1 (2)!
5C3 =
5∗4∗3
3∗2∗1
5C3 =
60
6
5C3 =
10
Therefore there are 10 possible samples of size 3 that can be drawn from the given data.
2. List all the possible samples and compute the mean of each sample.
Sample
2,49
2,4,10
2,4,5
2,9,10
2,9,5
2,10,5
̅)
Mean(𝒙
5
5.33
3.67
7
5.33
5.667
4,9,10
4,9,5
4,10,5
9,10,5
7.67
6
6.33
8
3. Construct the sampling distribution.
Sample Mean
3.67
5
5.33
5.67
6
6.33
7
7.67
8
Total
Frequency
1
1
2
1
1
1
1
1
1
10
Probability
1/10
1/10
2/10
1/10
1/10
1/10
1/10
1/10
1/10
10/10 or 1
4. Construct a histogram.
ACTIVITY: Construct the sampling distribution of the sample mean. (1
whole sheet of paper)
1. A certain population consist of the numbers 3, 6, 7, 12, and 4.
Construct a sampling distribution of size 3.
2. A population consists of the five numbers 1, 2, 3, 4, and 5. Consider
samples of size 3 that can be drawn from this population.