Certainly! Let's discuss how the values from your sample relate to a normally distributed population with a mean of 24 ounces and a standard deviation of 2 ounces.
### Overview of Expected Values
In a normally distributed population, important statistical measures such as the mean, median, and mode of the distribution are equal. This property holds true for populations that follow a normal distribution, making it easier to understand and make predictions about such populations.
### Sample Values
Here are the values from your sample:
- Sample Mean: 24 ounces
- Sample Standard Deviation: 2 ounces
### Population Values
The population under consideration has the following parameters:
- Population Mean (μ): 24 ounces
- Population Standard Deviation (σ): 2 ounces
### Comparison of Sample and Population Values
1. Mean:
- Population Mean: 24 ounces
- Sample Mean: 24 ounces
The sample mean is exactly equal to the population mean. This indicates that the average weight of the sample corresponds perfectly to the average weight of the entire population. In a normally distributed population, this result is expected when the sample size is reasonably large or when the sampling method is unbiased.
2. Standard Deviation:
- Population Standard Deviation: 2 ounces
- Sample Standard Deviation: 2 ounces
The sample standard deviation is also equal to the population standard deviation. This indicates that the spread, or variability, within your sample is consistent with the spread within the entire population. This consistency further supports the representativeness of the sample in relation to the population.
### Interpretation
From the given data, the sample values closely mirror the population parameters. This alignment suggests several points:
- Representativeness: The sample appears to be representative of the entire population, as both the mean and the standard deviation match the population values.
- Sampling Accuracy: It's likely that the sampling process was accurate and unbiased, capturing the true characteristics of the population.
- Central Limit Theorem: Even if the sample size is relatively small, the central limit theorem tells us that with random sampling, the sample mean, in particular, will tend to be normally distributed around the population mean. This principle further explains the alignment seen here.
### Conclusion
The mean and standard deviation of your sample are both equal to the respective population values (24 ounces for the mean and 2 ounces for the standard deviation). This indicates a high degree of accuracy in your sample, reflecting the characteristics of the normal distribution from which it was drawn. As expected, in a normally distributed population, these sample statistics provide a good estimation of the population parameters, especially when the sample size is reasonable.
