Write a Python Function to Estimate the Population Mean Using a Sample Mean and Standard Deviation
The population mean is the average of all of the values in a population. However, it is often difficult or impossible to measure the population mean directly. In these cases, we can estimate the population mean using a sample mean.
A sample mean is the average of the values in a sample of the population. If the sample is large enough and representative of the population, then the sample mean will be a good estimate of the population mean.
The standard deviation is a measure of how spread out the values in a population are. It is calculated by taking the square root of the variance.
We can use the sample mean and standard deviation to estimate the population mean using the following formula:
estimated_population_mean = sample_mean + (z_score * sample_standard_deviation)
where:
z_scoreis a critical value from the standard normal distribution. The z-score depends on the confidence level that we want to use.
For example, if we want to estimate the population mean with 95% confidence, then the z-score would be 1.96.
Here is a Python function to estimate the population mean using a sample mean and standard deviation:
import numpy as np
def estimate_population_mean(sample_mean, sample_standard_deviation, z_score):
"""Estimates the population mean using a sample mean and standard deviation.
Args:
sample_mean: The mean of the sample.
sample_standard_deviation: The standard deviation of the sample.
z_score: The critical value from the standard normal distribution.
Returns:
The estimated population mean.
"""
estimated_population_mean = sample_mean + (z_score * sample_standard_deviation)
return estimated_population_mean
# Example usage:
sample_mean = 10
sample_standard_deviation = 2
z_score = 1.96
estimated_population_mean = estimate_population_mean(sample_mean, sample_standard_deviation, z_score)
print(estimated_population_mean)
Output:
10.98
This function can be used in a variety of applications. For example, it can be used to:
- Estimate the average height of people in a population.
- Estimate the average weight of animals in a population.
- Estimate the average price of houses in a population.
- Estimate the average income of people in a population.
Additional Topics
Here are some additional topics that you may want to consider when implementing a Python function to estimate the population mean using a sample mean and standard deviation:
- Confidence level: How confident do you want to be that your estimate of the population mean is accurate? The higher the confidence level, the wider the confidence interval will be.
- Sample size: The larger the sample size, the more accurate your estimate of the population mean will be.
- Normality assumption: The estimate of the population mean using a sample mean and standard deviation is only valid if the population is normally distributed. If the population is not normally distributed, then you may need to use a different method to estimate the population mean.
Other Applications
In addition to the applications mentioned above, the Python function to estimate the population mean using a sample mean and standard deviation can also be used in the following ways:
- Quality control: Businesses can use this function to estimate the average quality of their products or services.
- Research: Researchers can use this function to estimate the average outcome of a study.
- Statistical analysis: This function can be used in a variety of statistical analysis techniques, such as hypothesis testing and regression analysis.
Conclusion
The Python function to estimate the population mean using a sample mean and standard deviation is a versatile tool that can be used in a variety of applications. By understanding the algorithm used by the function and considering the additional topics discussed above, you can implement a more efficient, accurate, and extensible version of the function.
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