The following is a histogram (right skewed) of the first ten terms of a geometric distribution with p = .4. The mean of this distribution is 2.5 and the standard deviation is approximately 1.93.
What's the shape of the distribution of sample means for simple random samples of size 5 drawn from this distribution? What are the mean and the standard deviation of the sampling distribution?
I am not sure if mean = p or sample mean = population mean
In this case, the shape of the distribution of sample means for simple random samples of size 5 drawn from this distribution is approximately normally distributed, according to the Central Limit Theorem. Regardless of the shape of the original distribution (in this case, right-skewed), when sample means are calculated from multiple samples drawn from the same population, the distribution of these sample means tends to approach a normal distribution.
To find the mean of the sampling distribution, you can use the formula:
mean of the sampling distribution = population mean = 2.5
The mean of the sampling distribution is equal to the population mean. This means that on average, the sample means will be equal to the population mean.
To find the standard deviation of the sampling distribution, you can use the formula:
standard deviation of the sampling distribution = standard deviation of the population / square root of the sample size
In this case, the standard deviation of the population is approximately 1.93, and the sample size is 5. Therefore, the standard deviation of the sampling distribution is:
standard deviation of the sampling distribution = 1.93 / √5 ≈ 0.864
So, the mean of the sampling distribution is 2.5 and the standard deviation is approximately 0.864.
To determine the shape of the distribution of sample means for simple random samples drawn from a geometric distribution, we need to consider the central limit theorem (CLT). The CLT states that for a large sample size (typically n > 30), the distribution of sample means will approach a normal distribution, regardless of the shape of the population distribution.
In this case, we have a geometric distribution with p = 0.4. Since the question specifies that the sample size is 5, which is relatively small, the CLT may not apply directly. However, we can make an approximation that the shape of the distribution of sample means will be approximately normal.
To find the mean (μ) and standard deviation (σ) of the sampling distribution, we use the following formulas:
μₓ = μ (population mean)
σₓ = σ/√n (population standard deviation divided by the square root of the sample size)
Given that the mean of the population distribution is 2.5 and the standard deviation is approximately 1.93, and using a sample size of 5, we can calculate the mean and standard deviation of the sampling distribution as follows:
μₓ = 2.5
σₓ = 1.93/√5 ≈ 0.865
Therefore, the shape of the distribution of sample means for simple random samples of size 5 drawn from this geometric distribution is approximately normal, with a mean of 2.5 and a standard deviation of approximately 0.865.