Question 1192946: 13. How many unique samples of size 5 are there in a population of 10 different numbers? 13.
14. Refer to problem 13 above. If you compared the population mean and the mean of the sampling
distribution of samples means (all sample means of size 5 used), what would you expect the difference
between the two numbers to be? Explain.
15. Refer to problem 13 above. If you randomly sampled 20 means from the sampling distribution of sample
means and found the average of those 20 means, how would you expect that value to compare to the
population mean? Explain.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **13. How many unique samples of size 5 are there in a population of 10 different numbers?**
* This is a combination problem: We want to choose 5 items from a set of 10, and the order of the items within a sample doesn't matter.
* The formula for combinations is:
* **nCr = n! / (r! * (n-r)!)**
* where:
* n is the total number of items (10 in this case)
* r is the number of items to choose (5 in this case)
* ! denotes factorial (e.g., 5! = 5 * 4 * 3 * 2 * 1)
* **Calculation:**
* 10C5 = 10! / (5! * (10-5)!)
* 10C5 = 10! / (5! * 5!)
* 10C5 = (10 * 9 * 8 * 7 * 6) / (5 * 4 * 3 * 2 * 1)
* 10C5 = 252
* **Therefore, there are 252 unique samples of size 5 in a population of 10 different numbers.**
**14. Refer to problem 13 above. If you compared the population mean and the mean of the sampling distribution of sample means (all sample means of size 5 used), what would you expect the difference between the two numbers to be? Explain.**
* **The expected difference between the population mean and the mean of the sampling distribution of sample means would be zero.**
* **Explanation:**
* The mean of the sampling distribution of sample means is also known as the **expected value of the sample mean**, and it is equal to the **population mean**.
* This is a fundamental concept in statistics.
**15. Refer to problem 13 above. If you randomly sampled 20 means from the sampling distribution of sample means and found the average of those 20 means, how would you expect that value to compare to the population mean? Explain.**
* **You would expect the average of those 20 means to be very close to the population mean.**
* **Explanation:**
* The sampling distribution of sample means is centered around the population mean.
* By taking a large enough sample of means (in this case, 20), you're essentially averaging out any random sampling fluctuations.
* This average of sample means will tend to be a very good estimate of the true population mean.
**Key Concepts:**
* **Sampling Distribution of the Mean:** The distribution of sample means that would result from drawing all possible samples of a given size from a population.
* **Central Limit Theorem:** This theorem states that as the sample size increases, the distribution of sample means approaches a normal distribution, regardless of the shape of the original population distribution.
* **Unbiased Estimator:** The mean of the sampling distribution of sample means is an unbiased estimator of the population mean.
I hope this explanation is helpful!
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