Final answer:
To compute the mean and variance of the sampling distribution of the sample means, we need to follow certain steps. First, compute the population mean and variance. Then, determine the number of possible samples and list all possible samples and their corresponding means. Next, construct the sampling distribution of the sample means and compute its mean and variance. Finally, construct the histogram.
Step-by-step explanation:
In order to complete the tasks, let's go step by step:
- Compute the population mean.
- Compute the population variance.
- Determine the number of possible samples.
- List all possible samples and their corresponding mean.
- Construct the sampling distribution of the sample means.
- Compute the mean of the sampling distribution of the sample means.
- Compute the variance of the sampling distribution of the sample means.
- Construct the Histogram.
1. To compute the population mean, add up all the numbers in the population (2+4+9+10+15) and divide by the total number of observations (5):
Population Mean = (2+4+9+10+15)/5 = 8.
2. To compute the population variance, we need to find the squared difference between each observation and the population mean, then add them up and divide by the total number of observations:
Population Variance = [(2-8)^2 + (4-8)^2 + (9-8)^2 + (10-8)^2 + (15-8)^2] / 5 = 17.6.
3. The number of possible samples can be calculated using the combination formula, where the number of items to choose from is the population size (5) and the sample size is the combination size (3):
Number of possible samples = 5 choose 3 = 10.
4. To list all possible samples and their corresponding mean, we can use the concept of combinations again. We can list all combinations of 3 numbers chosen from the population and calculate their mean:
(2, 4, 9) -> Mean = (2+4+9)/3 = 5
(2, 4, 10) -> Mean = (2+4+10)/3 = 5.333
(2, 4, 15) -> Mean = (2+4+15)/3 = 7
(2, 9, 10) -> Mean = (2+9+10)/3 = 7
(2, 9, 15) -> Mean = (2+9+15)/3 = 8.667
(2, 10, 15) -> Mean = (2+10+15)/3 = 9
(4, 9, 10) -> Mean = (4+9+10)/3 = 7.667
(4, 9, 15) -> Mean = (4+9+15)/3 = 9.333
(4, 10, 15) -> Mean = (4+10+15)/3 = 9.667
(9, 10, 15) -> Mean = (9+10+15)/3 = 11.333
5. The sampling distribution of the sample means can be constructed by listing all the means from step 4 and their corresponding frequencies:
Mean = 5, Frequency = 1
Mean = 5.333, Frequency = 1
Mean = 7, Frequency = 1
Mean = 7, Frequency = 1
Mean = 8.667, Frequency = 1
Mean = 9, Frequency = 1
Mean = 7.667, Frequency = 1
Mean = 9.333, Frequency = 1
Mean = 9.667, Frequency = 1
Mean = 11.333, Frequency = 1
6. The mean of the sampling distribution of the sample means can be calculated by adding up all the means and dividing by the total number of means (10):
Mean of the sampling distribution of the sample means = (5+5.333+7+7+8.667+9+7.667+9.333+9.667+11.333)/10 = 8.333.
7. The variance of the sampling distribution of the sample means can be calculated by finding the squared difference between each mean and the mean of the sampling distribution, then adding them up and dividing by the total number of means (10):
Variance of the sampling distribution of the sample means = [(5-8.333)^2 + (5.333-8.333)^2 + (7-8.333)^2 + (7-8.333)^2 + (8.667-8.333)^2 + (9-8.333)^2 + (7.667-8.333)^2 + (9.333-8.333)^2 + (9.667-8.333)^2 + (11.333-8.333)^2] / 10 = 1.745.
8. To construct the histogram, we can use the mean and standard deviation of the sampling distribution to determine the number of intervals and the width of each interval. The histogram will show the frequency of each mean from step 4.