The Margin of Error Formula

Oct 13, 2025 by leolavanderia.com.br 52 views

The margin of error is a crucial concept in statistics that helps us determine the reliability of a survey or experiment. It is the maximum amount by which the sample mean is expected to differ from the true population mean. In this article, we will explore the formula for calculating the margin of error and use it to determine which of the given options produce a margin of error of 0.95.

The Margin of Error Formula

The margin of error formula is given by:

${ M E = \frac{z \cdot s}{\sqrt{n}} }$

where:

$M E$ is the margin of error
$z$ is the z-score corresponding to the desired confidence level
$s$ is the standard deviation of the population
$n$ is the sample size

Confidence Level and Z-Score

The confidence level is the probability that the sample mean will be within the margin of error of the true population mean. A confidence level of 0.95 means that there is a 95% chance that the sample mean will be within the margin of error of the true population mean. The z-score corresponding to a confidence level of 0.95 is 1.96.

Given Options

We are given three options:

A. $z=2.14 ; s=4 ; n=9$ B. $z=2.14 ; s=4 ; n=81$ C. $z=1.96 ; s=4 ; n=81$

We need to determine which of these options produce a margin of error of 0.95.

Calculating the Margin of Error

Let's calculate the margin of error for each option:

Option A

${ M E = \frac{2.14 \cdot 4}{\sqrt{9}} }$ ${ M E = \frac{8.56}{3} }$ ${ M E = 2.8533 }$

Option B

${ M E = \frac{2.14 \cdot 4}{\sqrt{81}} }$ ${ M E = \frac{8.56}{9} }$ ${ M E = 0.9533 }$

Option C

${ M E = \frac{1.96 \cdot 4}{\sqrt{81}} }$ ${ M E = \frac{7.84}{9} }$ ${ M E = 0.8704 }$

Conclusion

Based on the calculations above, we can see that option B produces a margin of error of 0.95.

Margin of Error and Sample Size

The margin of error is inversely proportional to the square root of the sample size. This means that as the sample size increases, the margin of error decreases. In option B, the sample size is 81, which is much larger than the sample size in option A (9). This is why the margin of error in option B is much smaller than the margin of error in option A.

Margin of Error and Standard Deviation

The margin of error is also inversely proportional to the standard deviation of the population. This means that as the standard deviation increases, the margin of error decreases. In options A and B, the standard deviation is the same (4). Therefore, the margin of error is the same for both options.

Margin of Error and Z-Score

The margin of error is directly proportional to the z-score corresponding to the desired confidence level. This means that as the z-score increases, the margin of error increases. In options A and B, the z-score is the same (2.14). Therefore, the margin of error is the same for both options.

Conclusion

In conclusion, the margin of error is a crucial concept in statistics that helps us determine the reliability of a survey or experiment. The margin of error formula is given by:

${ M E = \frac{z \cdot s}{\sqrt{n}} }$

where:

$M E$ is the margin of error
$z$ is the z-score corresponding to the desired confidence level
$s$ is the standard deviation of the population
$n$ is the sample size

We have seen that the margin of error is inversely proportional to the square root of the sample size and inversely proportional to the standard deviation of the population. We have also seen that the margin of error is directly proportional to the z-score corresponding to the desired confidence level.

References

Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics (6th ed.). W.H. Freeman and Company.
Larson, R. E., & Farber, B. A. (2013). Elementary statistics: Picturing the world (5th ed.). Cengage Learning.

Discussion

The margin of error is a crucial concept in statistics that helps us determine the reliability of a survey or experiment. It is the maximum amount by which the sample mean is expected to differ from the true population mean. In this article, we have seen how to calculate the margin of error using the formula:

${ M E = \frac{z \cdot s}{\sqrt{n}} }$

We have also seen how the margin of error is affected by the sample size, standard deviation, and z-score corresponding to the desired confidence level.

What is the Margin of Error?

The margin of error is the maximum amount by which the sample mean is expected to differ from the true population mean.

What is the Formula for Calculating the Margin of Error?

The formula for calculating the margin of error is:

${ M E = \frac{z \cdot s}{\sqrt{n}} }$

where:

$M E$ is the margin of error
$z$ is the z-score corresponding to the desired confidence level
$s$ is the standard deviation of the population
$n$ is the sample size

How is the Margin of Error Affected by the Sample Size?

The margin of error is inversely proportional to the square root of the sample size. This means that as the sample size increases, the margin of error decreases.

How is the Margin of Error Affected by the Standard Deviation?

The margin of error is inversely proportional to the standard deviation of the population. This means that as the standard deviation increases, the margin of error decreases.

How is the Margin of Error Affected by the Z-Score?

The margin of error is directly proportional to the z-score corresponding to the desired confidence level. This means that as the z-score increases, the margin of error increases.

Conclusion

In conclusion, the margin of error is a crucial concept in statistics that helps us determine the reliability of a survey or experiment. The margin of error formula is given by:

${ M E = \frac{z \cdot s}{\sqrt{n}} }$

where:

$M E$ is the margin of error
$z$ is the z-score corresponding to the desired confidence level
$s$ is the standard deviation of the population
$n$ is the sample size

Q: What is the margin of error?

A: The margin of error is the maximum amount by which the sample mean is expected to differ from the true population mean.

Q: What is the formula for calculating the margin of error?

A: The formula for calculating the margin of error is:

${ M E = \frac{z \cdot s}{\sqrt{n}} }$

where:

$M E$ is the margin of error
$z$ is the z-score corresponding to the desired confidence level
$s$ is the standard deviation of the population
$n$ is the sample size

Q: How is the margin of error affected by the sample size?

A: The margin of error is inversely proportional to the square root of the sample size. This means that as the sample size increases, the margin of error decreases.

Q: How is the margin of error affected by the standard deviation?

A: The margin of error is inversely proportional to the standard deviation of the population. This means that as the standard deviation increases, the margin of error decreases.

Q: How is the margin of error affected by the z-score?

A: The margin of error is directly proportional to the z-score corresponding to the desired confidence level. This means that as the z-score increases, the margin of error increases.

Q: What is the z-score?

A: The z-score is a measure of how many standard deviations an observation is away from the mean. It is used to determine the confidence level of a survey or experiment.

Q: What is the confidence level?

A: The confidence level is the probability that the sample mean will be within the margin of error of the true population mean.

Q: What is the difference between a sample mean and a population mean?

A: The sample mean is the average of a sample of data, while the population mean is the average of the entire population.

Q: Why is it important to know the margin of error?

A: Knowing the margin of error is important because it helps us determine the reliability of a survey or experiment. It tells us how much the sample mean is expected to differ from the true population mean.

Q: How can I calculate the margin of error?

A: You can calculate the margin of error using the formula:

${ M E = \frac{z \cdot s}{\sqrt{n}} }$

where:

$M E$ is the margin of error
$z$ is the z-score corresponding to the desired confidence level
$s$ is the standard deviation of the population
$n$ is the sample size

Q: What is the relationship between the margin of error and the sample size?

A: The margin of error is inversely proportional to the square root of the sample size. This means that as the sample size increases, the margin of error decreases.

Q: What is the relationship between the margin of error and the standard deviation?

A: The margin of error is inversely proportional to the standard deviation of the population. This means that as the standard deviation increases, the margin of error decreases.

Q: What is the relationship between the margin of error and the z-score?

A: The margin of error is directly proportional to the z-score corresponding to the desired confidence level. This means that as the z-score increases, the margin of error increases.

Q: Can I use the margin of error to determine the reliability of a survey or experiment?

A: Yes, you can use the margin of error to determine the reliability of a survey or experiment. The margin of error tells us how much the sample mean is expected to differ from the true population mean.

Q: How can I use the margin of error to make decisions?

A: You can use the margin of error to make decisions by considering the following:

If the margin of error is small, it may indicate that the sample mean is a good estimate of the population mean.
If the margin of error is large, it may indicate that the sample mean is not a good estimate of the population mean.

Q: What are some common mistakes to avoid when calculating the margin of error?

A: Some common mistakes to avoid when calculating the margin of error include:

Using an incorrect z-score
Using an incorrect standard deviation
Using an incorrect sample size
Not considering the confidence level

Q: How can I improve my understanding of the margin of error?

A: You can improve your understanding of the margin of error by:

Practicing calculating the margin of error using different values
Reading about the margin of error in different contexts
Asking questions and seeking help when needed

Conclusion

In conclusion, the margin of error is a crucial concept in statistics that helps us determine the reliability of a survey or experiment. It is the maximum amount by which the sample mean is expected to differ from the true population mean. We have seen how to calculate the margin of error using the formula:

${ M E = \frac{z \cdot s}{\sqrt{n}} }$

We have also seen how the margin of error is affected by the sample size, standard deviation, and z-score corresponding to the desired confidence level. By understanding the margin of error, we can make more informed decisions and improve the reliability of our surveys and experiments.