The sample mean to be calculated from a random sample of size n= 4 from a population consistsof eight measurements (2, 6, 9, 12, 25, 29, 39, 50) . find the sampling distribution of y y .(hint : there are 70 samplesof size 4 when sampling from a population of eight measurements )
To find the sampling distribution of the sample mean (y), we need to calculate the sample means for all possible samples of size 4 that can be taken from the population. Here's how to do it step by step:
1. Determine the number of possible samples: In this case, we have 8 measurements in the population, and we want to select samples of size 4. The formula to calculate the number of possible samples is given by nCr, where n is the number of measurements in the population and r is the sample size. So, nCr = 8C4 = (8!)/(4!(8-4)!) = 70. Hence, there are 70 possible samples of size 4 that can be taken.
2. List all the possible samples: To find the sampling distribution, we need to find the mean of each of the 70 possible samples. Let's list all the possible samples from the given measurements:
- Sample 1: (2, 6, 9, 12)
- Sample 2: (2, 6, 9, 25)
- Sample 3: (2, 6, 9, 29)
- ...
- Sample 70: (12, 25, 29, 50)
3. Calculate the mean of each sample: For each sample, calculate the mean by summing up all the measurements in the sample and dividing it by the sample size (which is 4 in this case).
- Sample 1 mean = (2 + 6 + 9 + 12)/4 = 29/4
- Sample 2 mean = (2 + 6 + 9 + 25)/4 = 42/4
- Sample 3 mean = (2 + 6 + 9 + 29)/4 = 46/4
- ...
- Sample 70 mean = (12 + 25 + 29 + 50)/4 = 116/4
4. Record the sample means: After calculating the mean for each sample, record the values in a list. This list of means represents the sampling distribution of the sample mean (y).
The sampling distribution of y will have 70 values in this case.
I hope this explanation helps you understand how to find the sampling distribution of the sample mean.