Question

Assume the weekly costs for full-time daycare in some large city have a mean of $120, and a standard deviation of $24. Suppose a random sample of 40 daycare facilities is selected.Give the properties of the sampling distribution for the sample mean weekly cost.

Answer 1

The properties of the sampling distribution for the sample mean weekly cost can be described as follows: the mean of the sampling distribution is equal to the population mean, and the standard deviation of the sampling distribution can be calculated using the formula: Standard Error = (Standard Deviation of Population) / sqrt(sample size).

Step-by-step explanation:

The properties of the sampling distribution for the sample mean weekly cost can be described as follows:

1. The mean of the sampling distribution will be equal to the population mean, which is $120 in this case.
2. The standard deviation of the sampling distribution, also known as the standard error of the mean, can be calculated using the formula:
3. Standard Error = (Standard Deviation of Population) / sqrt(sample size)
4. Substituting the values, we get:
5. Standard Error = $24 / sqrt(40)
6. Standard Error ≈ $3.79

Answer 2

The sampling distribution of the sample mean weekly cost of daycare facilities will have a mean of $120 and a standard deviation of $3.80.

Step-by-step explanation:

The sampling distribution for the sample mean is characterized by a mean that remains the same as the population mean, which is $120 in this case. This represents the center of the distribution of sample means.

Additionally, the standard deviation of the sampling distribution of the sample mean, known as the standard error (SE), is calculated by dividing the population standard deviation by the square root of the sample size.

Here, with a population standard deviation of $24 and a sample size of 40, the standard error would be $24 / √40 ≈ $3.80. The standard error measures the variability of sample means around the population mean.

In essence, as the sample size increases, the standard error decreases, indicating that larger sample sizes tend to provide a more accurate estimate of the population mean.

Consequently, the sampling distribution becomes more concentrated around the population mean, illustrating the concept of the Central Limit Theorem, which states that as the sample size increases, the sampling distribution of the sample mean becomes approximately normally distributed, regardless of the population distribution.