A Random Sample Of N = 25 N = 25 N = 25 Scores Is Obtained From A Population With Μ = 40 \mu = 40 Μ = 40 And Σ = 8 \sigma = 8 Σ = 8 . If The Sample Mean Is M = 36 M = 36 M = 36 , What Is The Z Z Z -score Corresponding To This Sample Mean?

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Understanding the Problem

When working with a sample of data, it's often necessary to determine how the sample mean compares to the population mean. This can be done by calculating the z-score, which measures the number of standard deviations a sample mean is away from the population mean. In this scenario, we have a sample of $n = 25$ scores from a population with a mean of $\mu = 40$ and a standard deviation of $\sigma = 8$. The sample mean is $M = 36$, and we want to find the z-score corresponding to this sample mean.

Calculating the z-Score

The formula for calculating the z-score is:

$$z = \frac{M - \mu}{\frac{\sigma}{\sqrt{n}}}$$

Where:

• $M$ is the sample mean
• $\mu$ is the population mean
• $\sigma$ is the population standard deviation
• $n$ is the sample size

In this case, we have:

• $M = 36$
• $\mu = 40$
• $\sigma = 8$
• $n = 25$

Plugging these values into the formula, we get:

$$z = \frac{36 - 40}{\frac{8}{\sqrt{25}}}$$

$$z = \frac{-4}{\frac{8}{5}}$$

$$z = \frac{-4}{1.6}$$

$$z = -2.5$$

Interpreting the z-Score

The z-score of -2.5 indicates that the sample mean of 36 is 2.5 standard deviations below the population mean of 40. This means that the sample mean is significantly lower than the population mean, suggesting that the sample may not be representative of the population.

Understanding the Significance of the z-Score

The z-score is a measure of how many standard deviations away from the population mean a sample mean is. A z-score of 0 indicates that the sample mean is equal to the population mean, while a positive z-score indicates that the sample mean is above the population mean, and a negative z-score indicates that the sample mean is below the population mean.

Using the z-Score to Make Inferences

The z-score can be used to make inferences about the population based on the sample data. For example, if the z-score is within a certain range (e.g. -1 to 1), it may indicate that the sample is representative of the population. However, if the z-score is outside of this range, it may indicate that the sample is not representative of the population.

Conclusion

In conclusion, the z-score is a useful tool for determining how a sample mean compares to the population mean. By calculating the z-score, we can determine how many standard deviations away from the population mean a sample mean is. In this scenario, the z-score of -2.5 indicates that the sample mean of 36 is 2.5 standard deviations below the population mean of 40.

Example Use Case

Suppose we have a sample of exam scores from a class of 25 students, and we want to determine how the sample mean compares to the population mean. We calculate the sample mean to be 80, and the population mean to be 85. We also know that the population standard deviation is 10. Using the formula for the z-score, we get:

$$z = \frac{80 - 85}{\frac{10}{\sqrt{25}}}$$

$$z = \frac{-5}{\frac{10}{5}}$$

$$z = \frac{-5}{2}$$

$$z = -2.5$$

This indicates that the sample mean of 80 is 2.5 standard deviations below the population mean of 85.

References

• [1] Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
• [2] Larson, R. E., & Farber, B. A. (2018). Elementary statistics: Picturing the world. Cengage Learning.

Discussion

The z-score is a useful tool for determining how a sample mean compares to the population mean. However, it's essential to consider the limitations of the z-score, such as the assumption of normality and the potential for outliers. Additionally, the z-score should be used in conjunction with other statistical methods to make informed inferences about the population.

Final Thoughts

In conclusion, the z-score is a powerful tool for analyzing sample data and making inferences about the population. By understanding how to calculate and interpret the z-score, researchers and practitioners can gain valuable insights into the characteristics of their sample data and make more informed decisions.

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Q&A: Understanding the z-Score

Q: What is the z-score, and why is it important?

A: The z-score is a measure of how many standard deviations away from the population mean a sample mean is. It's essential to calculate the z-score because it helps us understand how representative the sample is of the population.

Q: How do I calculate the z-score?

A: To calculate the z-score, you need to use the following formula:

$$z = \frac{M - \mu}{\frac{\sigma}{\sqrt{n}}}$$

Where:

• $M$ is the sample mean
• $\mu$ is the population mean
• $\sigma$ is the population standard deviation
• $n$ is the sample size

Q: What does a positive z-score indicate?

A: A positive z-score indicates that the sample mean is above the population mean. This means that the sample is likely to be representative of the population.

Q: What does a negative z-score indicate?

A: A negative z-score indicates that the sample mean is below the population mean. This means that the sample may not be representative of the population.

Q: How do I interpret the z-score?

A: The z-score can be interpreted as follows:

• A z-score of 0 indicates that the sample mean is equal to the population mean.
• A positive z-score indicates that the sample mean is above the population mean.
• A negative z-score indicates that the sample mean is below the population mean.

Q: What is the significance of the z-score in hypothesis testing?

A: The z-score is used in hypothesis testing to determine whether the sample mean is significantly different from the population mean. If the z-score is within a certain range (e.g. -1 to 1), it may indicate that the sample is representative of the population.

Q: Can I use the z-score to make inferences about the population?

A: Yes, the z-score can be used to make inferences about the population based on the sample data. However, it's essential to consider the limitations of the z-score, such as the assumption of normality and the potential for outliers.

Q: What are some common applications of the z-score?

A: The z-score is commonly used in:

• Hypothesis testing
• Confidence intervals
• Regression analysis
• Time series analysis

Q: Can I use the z-score with non-normal data?

A: No, the z-score assumes normality of the data. If the data is not normally distributed, you may need to use alternative methods, such as the t-test or non-parametric tests.

Q: How do I choose between the z-score and the t-test?

A: The z-score is used when the sample size is large (n > 30), and the population standard deviation is known. The t-test is used when the sample size is small (n < 30), or the population standard deviation is unknown.

Q: What are some common mistakes to avoid when using the z-score?

A: Some common mistakes to avoid when using the z-score include:

• Assuming normality of the data
• Ignoring outliers
• Using the z-score with small sample sizes
• Not considering the limitations of the z-score

Conclusion

In conclusion, the z-score is a powerful tool for analyzing sample data and making inferences about the population. By understanding how to calculate and interpret the z-score, researchers and practitioners can gain valuable insights into the characteristics of their sample data and make more informed decisions.