A Normal Population Has A Mean Μ = 40 \mu = 40 Μ = 40 And A Standard Deviation Of Σ = 8 \sigma = 8 Σ = 8 . After A Treatment Is Administered To A Sample Of N = 16 N = 16 N = 16 Individuals From The Population, The Sample Mean Is Found To Be $M =

Mar 12, 2025 by ADMIN 246 views

A Normal Population: Understanding the Impact of Treatment on Sample Mean

Introduction

In statistics, understanding the behavior of a population and its characteristics is crucial for making informed decisions. A normal population is characterized by a mean ( $\mu$ ) and a standard deviation ( $\sigma$ ). In this article, we will explore the impact of a treatment on the sample mean of a normal population. We will use the given information: a normal population has a mean $\mu = 40$ and a standard deviation of $\sigma = 8$ . A sample of $n = 16$ individuals from the population is taken, and the sample mean is found to be $M = 42$ . Our goal is to understand the significance of this change in the sample mean.

Understanding the Normal Distribution

A normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In a normal distribution, about 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and about 99.7% falls within three standard deviations. This distribution is often referred to as the "bell curve" due to its shape.

The 68-95-99.7 Rule

About 68% of the data falls within one standard deviation of the mean ( $\mu \pm \sigma$ ).
About 95% of the data falls within two standard deviations of the mean ( $\mu \pm 2\sigma$ ).
About 99.7% of the data falls within three standard deviations of the mean ( $\mu \pm 3\sigma$ ).

Calculating the Standard Error of the Mean

The standard error of the mean (SEM) is a measure of the variability of the sample mean. It is calculated by dividing the population standard deviation ( $\sigma$ ) by the square root of the sample size ( $n$ ). In this case, the population standard deviation is $\sigma = 8$ and the sample size is $n = 16$ . Therefore, the standard error of the mean is:

SEM = \frac{\sigma}{\sqrt{n}} = \frac{8}{\sqrt{16}} = \frac{8}{4} = 2

Understanding the Impact of Treatment on Sample Mean

The sample mean ( $M$ ) is a measure of the average value of the sample. In this case, the sample mean is $M = 42$ . To understand the impact of the treatment on the sample mean, we need to calculate the z-score. The z-score is a measure of how many standard errors away from the population mean the sample mean is.

z = \frac{M - \mu}{SEM} = \frac{42 - 40}{2} = \frac{2}{2} = 1

The z-score of 1 indicates that the sample mean is 1 standard error away from the population mean. This suggests that the treatment has had a significant impact on the sample mean.

Interpreting the Results

The results suggest that the treatment has had a significant impact on the sample mean. The sample mean is 1 standard error away from the population mean, indicating that the treatment has resulted in a significant change in the sample mean. This change is statistically significant, as it is greater than 1 standard error.

Conclusion

In conclusion, the treatment has had a significant impact on the sample mean of the normal population. The sample mean is 1 standard error away from the population mean, indicating that the treatment has resulted in a significant change in the sample mean. This change is statistically significant, as it is greater than 1 standard error. The standard error of the mean is a useful measure of the variability of the sample mean, and it can be used to calculate the z-score, which is a measure of how many standard errors away from the population mean the sample mean is.

Future Research Directions

Future research directions could include:

Investigating the long-term effects of the treatment on the sample mean.
Examining the relationship between the treatment and other variables, such as age and sex.
Replicating the study to confirm the findings.

Limitations of the Study

The study has several limitations, including:

The sample size is relatively small, which may limit the generalizability of the findings.
The study only examines the impact of the treatment on the sample mean, and does not examine other variables, such as the standard deviation.

References

[1] Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
[2] Rosner, B. (2010). Fundamentals of biostatistics. Cengage Learning.

Glossary

Normal distribution: A probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.
Standard error of the mean: A measure of the variability of the sample mean.
Z-score: A measure of how many standard errors away from the population mean the sample mean is.

Introduction

In our previous article, we explored the impact of a treatment on the sample mean of a normal population. We used the given information: a normal population has a mean $\mu = 40$ and a standard deviation of $\sigma = 8$ . A sample of $n = 16$ individuals from the population is taken, and the sample mean is found to be $M = 42$ . In this article, we will answer some frequently asked questions (FAQs) related to the topic.

Q&A

Q1: What is the standard error of the mean (SEM)?

A1: The standard error of the mean (SEM) is a measure of the variability of the sample mean. It is calculated by dividing the population standard deviation ( $\sigma$ ) by the square root of the sample size ( $n$ ). In this case, the population standard deviation is $\sigma = 8$ and the sample size is $n = 16$ . Therefore, the standard error of the mean is:

SEM = \frac{\sigma}{\sqrt{n}} = \frac{8}{\sqrt{16}} = \frac{8}{4} = 2

Q2: What is the z-score?

A2: The z-score is a measure of how many standard errors away from the population mean the sample mean is. It is calculated by dividing the difference between the sample mean ( $M$ ) and the population mean ( $\mu$ ) by the standard error of the mean (SEM). In this case, the z-score is:

z = \frac{M - \mu}{SEM} = \frac{42 - 40}{2} = \frac{2}{2} = 1

Q3: What does the z-score of 1 indicate?

A3: The z-score of 1 indicates that the sample mean is 1 standard error away from the population mean. This suggests that the treatment has had a significant impact on the sample mean.

Q4: What is the significance of the z-score?

A4: The z-score is a measure of the statistical significance of the sample mean. A z-score of 1 or greater indicates that the sample mean is statistically significant, meaning that it is unlikely to occur by chance.

Q5: What are the limitations of the study?

A5: The study has several limitations, including:

The sample size is relatively small, which may limit the generalizability of the findings.
The study only examines the impact of the treatment on the sample mean, and does not examine other variables, such as the standard deviation.

Q6: What are some future research directions?

A6: Some future research directions could include:

Investigating the long-term effects of the treatment on the sample mean.
Examining the relationship between the treatment and other variables, such as age and sex.
Replicating the study to confirm the findings.

Conclusion

Glossary

Normal distribution: A probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.
Standard error of the mean: A measure of the variability of the sample mean.
Z-score: A measure of how many standard errors away from the population mean the sample mean is.
Statistical significance: A measure of the likelihood that a result occurred by chance.

References

[1] Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
[2] Rosner, B. (2010). Fundamentals of biostatistics. Cengage Learning.