Given A Population With A Mean Of 76 And A Variance Of 256, What Is The Probability That The Mean Of A Random Sample Of 100 Will Be Between 75 And 78?

Jun 20, 2025 by ADMIN 151 views

Understanding the Probability of Sample Mean: A Detailed Analysis

In the realm of statistics, understanding the behavior of sample means is crucial for making inferences about populations. This article delves into a specific scenario where we explore the probability of a sample mean falling within a certain range. We'll dissect the problem, apply relevant statistical concepts, and arrive at a solution, providing a comprehensive understanding of the underlying principles. This article aims to clarify the process of calculating the probability of a sample mean falling within a specified range, a fundamental concept in inferential statistics. By understanding these principles, we can make informed decisions based on sample data and draw meaningful conclusions about the larger population from which the sample was drawn. The application of the Central Limit Theorem is key to solving this problem, as it allows us to approximate the distribution of the sample mean even when the population distribution is unknown.

Problem Statement

We are given a scenario where a random sample of size 100 is drawn from an infinite population. This population has a known mean (μ) of 76 and a variance (σ^2) of 256. Our objective is to determine the probability that the sample mean (β) will fall between 75 and 78. This is a classic problem in statistical inference, requiring the application of key concepts like the Central Limit Theorem and the calculation of z-scores. To accurately solve this problem, we need to first understand the distribution of the sample mean. The Central Limit Theorem (CLT) plays a pivotal role here, as it allows us to approximate this distribution under certain conditions. Before diving into the calculations, let's first recap the core principles that govern this type of probability problem, ensuring we have a solid foundation for our analysis.

Core Concepts

Central Limit Theorem (CLT)

The Central Limit Theorem (CLT) is a cornerstone of statistics. It states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the population's distribution. This theorem is crucial because it allows us to make inferences about population parameters even when we don't know the population's distribution. The Central Limit Theorem is applicable when the sample size is sufficiently large, generally considered to be 30 or more. The CLT is not just a theoretical construct; it has profound practical implications. It allows us to use the well-understood properties of the normal distribution to make inferences about population parameters based on sample statistics. This is particularly useful in situations where collecting data from the entire population is infeasible or impossible. Furthermore, the CLT provides a foundation for many statistical tests and confidence intervals, making it an indispensable tool in data analysis.

Standard Error of the Mean

The standard error of the mean (SEM) measures the variability of sample means around the population mean. It's calculated as the population standard deviation (σ) divided by the square root of the sample size (n): SEM = σ / √n. The standard error of the mean is a crucial concept in inferential statistics, as it provides a measure of the precision of the sample mean as an estimator of the population mean. A smaller standard error indicates that the sample means are clustered more tightly around the population mean, suggesting a more reliable estimate. Conversely, a larger standard error suggests greater variability in the sample means, which may lead to less precise estimates. Understanding the standard error is essential for constructing confidence intervals and conducting hypothesis tests, as it forms the basis for calculating test statistics and p-values. It is also important to note that the standard error decreases as the sample size increases, reflecting the fact that larger samples provide more accurate estimates of the population mean.

Z-score

A z-score quantifies how many standard deviations a data point is from the mean. For sample means, the z-score is calculated as: z = (β - μ) / SEM, where β is the sample mean, μ is the population mean, and SEM is the standard error of the mean. The z-score is a standardized measure that allows us to compare data points from different distributions. By converting values to z-scores, we can assess their relative positions within their respective distributions and determine their probabilities using the standard normal distribution table or statistical software. The z-score is particularly useful in hypothesis testing, where it is used to determine the statistical significance of a sample mean. It also plays a critical role in constructing confidence intervals, as it helps to define the range of values within which the population mean is likely to fall. Moreover, the z-score is a versatile tool for identifying outliers in a dataset and for comparing data points across different scales and units of measurement.

Calculation Steps

1. Calculate the Standard Error of the Mean (SEM)

Given the population variance (σ^2) is 256, the population standard deviation (σ) is the square root of 256, which is 16. With a sample size (n) of 100, the standard error of the mean (SEM) is calculated as follows:

SEM = σ / √n = 16 / √100 = 16 / 10 = 1.6

This step is crucial because the standard error forms the basis for calculating the z-scores, which are necessary for determining the probabilities. The standard error reflects the variability of the sample means around the population mean. A smaller standard error indicates that the sample means are more likely to be close to the population mean, while a larger standard error suggests greater variability. By accurately calculating the standard error, we can proceed with confidence in our subsequent calculations and interpretations. The standard error also plays a vital role in determining the width of confidence intervals, providing a measure of the uncertainty associated with our estimate of the population mean. Therefore, a thorough understanding of the standard error is essential for making informed decisions based on sample data.

2. Calculate the Z-scores

We need to calculate the z-scores for both boundaries of the interval (75 and 78). The formula for the z-score is:

z = (β - μ) / SEM

For β = 75: z1 = (75 - 76) / 1.6 = -1 / 1.6 = -0.625

For β = 78: z2 = (78 - 76) / 1.6 = 2 / 1.6 = 1.25

The z-scores provide a standardized measure of how far the sample means (75 and 78) are from the population mean (76) in terms of standard errors. A negative z-score indicates that the sample mean is below the population mean, while a positive z-score indicates that it is above. The magnitude of the z-score reflects the distance from the mean in standard deviation units. These calculated z-scores are essential for determining the probabilities associated with the sample mean falling within the specified range. By converting the sample means to z-scores, we can utilize the standard normal distribution table or statistical software to find the corresponding probabilities. The accuracy of these z-score calculations is paramount, as they directly impact the final probability estimate. Therefore, careful attention to detail is necessary to ensure the reliability of the results.

3. Find the Probabilities Using the Standard Normal Distribution

Using a standard normal distribution table or a statistical calculator, we find the probabilities corresponding to the calculated z-scores:

P(z < -0.625) ≈ 0.2660 P(z < 1.25) ≈ 0.8944

These probabilities represent the area under the standard normal curve to the left of the respective z-scores. The probability P(z < -0.625) indicates the likelihood of observing a sample mean less than 75, while P(z < 1.25) represents the likelihood of observing a sample mean less than 78. To find the probability of the sample mean falling between 75 and 78, we need to calculate the difference between these two probabilities. This step is crucial because it allows us to quantify the likelihood of the sample mean falling within the specified range, providing valuable information for statistical inference. The accuracy of these probability values is dependent on the precision of the standard normal distribution table or the statistical software used. Therefore, it is important to use reliable resources and tools to obtain accurate probability estimates.

4. Calculate the Probability

To find the probability that β is between 75 and 78, we subtract the probability of z being less than -0.625 from the probability of z being less than 1.25:

P(75 < β < 78) = P(-0.625 < z < 1.25) = P(z < 1.25) - P(z < -0.625) ≈ 0.8944 - 0.2660 = 0.6284

Therefore, the probability that the sample mean (β) will be between 75 and 78 is approximately 0.6284 or 62.84%. This final calculation provides the answer to our original problem statement. It quantifies the likelihood of observing a sample mean within the specified range, given the population parameters and the sample size. This probability can be used to make inferences about the population mean based on the sample data. The accuracy of this probability estimate depends on the correctness of all previous calculations, including the standard error, z-scores, and probabilities from the standard normal distribution. Therefore, it is essential to review each step carefully to ensure the reliability of the final result. This calculated probability serves as a valuable tool for decision-making and hypothesis testing in statistical analysis.

Conclusion

In conclusion, the probability that the sample mean (β) will be between 75 and 78 is approximately 0.6284. This result was obtained by applying the Central Limit Theorem, calculating the standard error of the mean, determining the z-scores for the interval boundaries, and using the standard normal distribution to find the corresponding probabilities. This problem illustrates the power of statistical inference in making predictions about population parameters based on sample data. The Central Limit Theorem is a fundamental concept in statistics, allowing us to approximate the distribution of sample means even when the population distribution is unknown. The standard error of the mean provides a measure of the variability of sample means around the population mean, and z-scores standardize the values, making it possible to use the standard normal distribution for probability calculations. By carefully applying these concepts, we can effectively analyze sample data and draw meaningful conclusions about the population from which the sample was drawn. This process is essential for a wide range of applications, from scientific research to business decision-making, highlighting the importance of statistical literacy in today's data-driven world.

This comprehensive analysis demonstrates the steps involved in calculating the probability of a sample mean falling within a specific range. By understanding these concepts, you can apply them to various statistical problems and make informed decisions based on data.