Describe the distribution of sample means (shape,
expected value, and standard error) for samples of
n � 36 selected from a population with a mean of
μ � 100 and a standard deviation of � � 12.
If it is a normal distribution, I would expect mean = 100.
SEm = SD/√n
insert values and calculate.
To describe the distribution of sample means, we need to consider three characteristics: shape, expected value, and standard error.
1. Shape:
The shape of the distribution of sample means is described by the Central Limit Theorem (CLT). According to the CLT, regardless of the shape of the population, as the sample size (n) increases, the distribution of sample means approaches a normal distribution. In this case, since the sample size (n) is 36, we can assume that the distribution of sample means will be approximately normal.
2. Expected Value (mean of sample means):
The expected value of the sample means (also known as the mean of the sampling distribution) is equal to the population mean (μ). In this case, the population mean (μ) is 100. Therefore, the expected value of the sample means is also 100.
3. Standard Error:
The standard error of the sample means (SE) is calculated using the formula: SE = σ / sqrt(n), where σ is the population standard deviation and n is the sample size. In this case, the population standard deviation (σ) is 12 and the sample size (n) is 36. Plugging these values into the formula, we get:
SE = 12 / sqrt(36) = 12 / 6 = 2
Therefore, the standard error of the sample means is 2.
In summary, the distribution of sample means, for samples of n = 36 selected from a population with a mean of μ = 100 and a standard deviation of σ = 12, will have an approximately normal shape. The expected value (mean of sample means) will be 100, and the standard error will be 2.