Among 320 Randomly Selected Airline Travelers, The Mean Number Of Hours Spent Traveling Per Year Is 24 Hours, And The Standard Deviation Is 2.9.What Is The Margin Of Error, Assuming A $90\%$ Confidence Level? Round Your Answer To The Nearest

Introduction

In statistical analysis, the margin of error is a crucial concept that helps us understand the reliability of our estimates. It represents the maximum amount by which a sample statistic may differ from the true population parameter. In this article, we will explore the concept of margin of error, its importance, and how to calculate it using a real-world example.

What is Margin of Error?

The margin of error is a measure of the uncertainty associated with a sample statistic. It is calculated as the product of the critical value from the standard normal distribution, the standard deviation of the population, and the square root of the sample size. The margin of error is usually expressed as a percentage of the sample statistic.

Importance of Margin of Error

The margin of error is essential in statistical analysis because it helps us:

• Interpret results: By understanding the margin of error, we can interpret the results of our analysis and make informed decisions.
• Compare estimates: The margin of error allows us to compare estimates from different samples and determine which one is more reliable.
• Make predictions: By calculating the margin of error, we can make predictions about the population parameter with a certain level of confidence.

Calculating Margin of Error

To calculate the margin of error, we need to follow these steps:

1. Determine the confidence level: The confidence level is the probability that the sample statistic will be within the margin of error. In this example, we are assuming a 90% confidence level.
2. Calculate the critical value: The critical value is the value from the standard normal distribution that corresponds to the desired confidence level. For a 90% confidence level, the critical value is approximately 1.645.
3. Calculate the standard error: The standard error is the standard deviation of the population divided by the square root of the sample size. In this example, the standard deviation is 2.9, and the sample size is 320.
4. Calculate the margin of error: The margin of error is the product of the critical value, the standard error, and the sample statistic.

Example: Calculating Margin of Error

Let's use the example given in the problem statement to calculate the margin of error.

• Sample statistic: The mean number of hours spent traveling per year is 24 hours.
• Standard deviation: The standard deviation is 2.9.
• Sample size: The sample size is 320.
• Confidence level: The confidence level is 90%.

First, we need to calculate the standard error:

Standard Error

$$\text{Standard Error} = \frac{\text{Standard Deviation}}{\sqrt{\text{Sample Size}}}$$

$$\text{Standard Error} = \frac{2.9}{\sqrt{320}}$$

$$\text{Standard Error} = \frac{2.9}{17.89}$$

$$\text{Standard Error} = 0.162$$

Next, we need to calculate the margin of error:

Margin of Error

$$\text{Margin of Error} = \text{Critical Value} \times \text{Standard Error} \times \text{Sample Statistic}$$

$$\text{Margin of Error} = 1.645 \times 0.162 \times 24$$

$$\text{Margin of Error} = 7.93$$

Therefore, the margin of error is approximately 7.93 hours.

Conclusion

In conclusion, the margin of error is a crucial concept in statistical analysis that helps us understand the reliability of our estimates. By calculating the margin of error, we can interpret results, compare estimates, and make predictions about the population parameter with a certain level of confidence. In this article, we used a real-world example to demonstrate how to calculate the margin of error using a 90% confidence level.

Margin of Error Formula

The margin of error formula is:

$$\text{Margin of Error} = \text{Critical Value} \times \text{Standard Error} \times \text{Sample Statistic}$$

Where:

• Critical Value: The value from the standard normal distribution that corresponds to the desired confidence level.
• Standard Error: The standard deviation of the population divided by the square root of the sample size.
• Sample Statistic: The sample mean or proportion.

Margin of Error Calculator

To calculate the margin of error, you can use the following calculator:

• Confidence Level: Enter the desired confidence level (e.g., 90%).
• Standard Deviation: Enter the standard deviation of the population.
• Sample Size: Enter the sample size.
• Sample Statistic: Enter the sample mean or proportion.
• Critical Value: The calculator will calculate the critical value based on the desired confidence level.
• Standard Error: The calculator will calculate the standard error based on the standard deviation and sample size.
• Margin of Error: The calculator will calculate the margin of error based on the critical value, standard error, and sample statistic.

Margin of Error Example

Let's use the example given in the problem statement to calculate the margin of error.

• Sample Statistic: The mean number of hours spent traveling per year is 24 hours.
• Standard Deviation: The standard deviation is 2.9.
• Sample Size: The sample size is 320.
• Confidence Level: The confidence level is 90%.

Using the calculator, we get:

• Critical Value: 1.645
• Standard Error: 0.162
• Margin of Error: 7.93

Therefore, the margin of error is approximately 7.93 hours.

Margin of Error Interpretation

The margin of error represents the maximum amount by which a sample statistic may differ from the true population parameter. In this example, the margin of error is approximately 7.93 hours, which means that the true mean number of hours spent traveling per year may be as high as 31.93 hours or as low as 16.07 hours.

Margin of Error Limitations

The margin of error has some limitations:

• Assumes normal distribution: The margin of error assumes that the population is normally distributed, which may not always be the case.
• Requires large sample size: The margin of error requires a large sample size to be accurate.
• Does not account for bias: The margin of error does not account for bias in the sample.

Margin of Error Conclusion

Q: What is the margin of error?

A: The margin of error is a measure of the uncertainty associated with a sample statistic. It represents the maximum amount by which a sample statistic may differ from the true population parameter.

Q: Why is the margin of error important?

A: The margin of error is essential in statistical analysis because it helps us:

• Interpret results: By understanding the margin of error, we can interpret the results of our analysis and make informed decisions.
• Compare estimates: The margin of error allows us to compare estimates from different samples and determine which one is more reliable.
• Make predictions: By calculating the margin of error, we can make predictions about the population parameter with a certain level of confidence.

Q: How do I calculate the margin of error?

A: To calculate the margin of error, you need to follow these steps:

1. Determine the confidence level: The confidence level is the probability that the sample statistic will be within the margin of error. In this example, we are assuming a 90% confidence level.
2. Calculate the critical value: The critical value is the value from the standard normal distribution that corresponds to the desired confidence level. For a 90% confidence level, the critical value is approximately 1.645.
3. Calculate the standard error: The standard error is the standard deviation of the population divided by the square root of the sample size. In this example, the standard deviation is 2.9, and the sample size is 320.
4. Calculate the margin of error: The margin of error is the product of the critical value, the standard error, and the sample statistic.

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

A: The margin of error formula is:

$$\text{Margin of Error} = \text{Critical Value} \times \text{Standard Error} \times \text{Sample Statistic}$$

Where:

• Critical Value: The value from the standard normal distribution that corresponds to the desired confidence level.
• Standard Error: The standard deviation of the population divided by the square root of the sample size.
• Sample Statistic: The sample mean or proportion.

Q: How do I choose the confidence level?

A: The confidence level is the probability that the sample statistic will be within the margin of error. Common confidence levels include:

• 90%: This is a common confidence level used in many statistical analyses.
• 95%: This is a higher confidence level that provides more accurate results.
• 99%: This is an even higher confidence level that provides even more accurate results.

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

A: The margin of error and the standard error are related but distinct concepts.

• Standard Error: The standard error is the standard deviation of the population divided by the square root of the sample size. It represents the amount of variation in the sample statistic.
• Margin of Error: The margin of error is the product of the critical value, the standard error, and the sample statistic. It represents the maximum amount by which a sample statistic may differ from the true population parameter.

Q: Can I use the margin of error to make predictions about the population parameter?

A: Yes, you can use the margin of error to make predictions about the population parameter with a certain level of confidence. However, you need to be aware of the limitations of the margin of error, including:

• Assumes normal distribution: The margin of error assumes that the population is normally distributed, which may not always be the case.
• Requires large sample size: The margin of error requires a large sample size to be accurate.
• Does not account for bias: The margin of error does not account for bias in the sample.

Q: How do I interpret the margin of error?

A: The margin of error represents the maximum amount by which a sample statistic may differ from the true population parameter. For example, if the margin of error is 5, it means that the true population parameter may be as high as 5 above the sample statistic or as low as 5 below the sample statistic.

Q: Can I use the margin of error to compare estimates from different samples?

A: Yes, you can use the margin of error to compare estimates from different samples. By calculating the margin of error for each sample, you can determine which one is more reliable.

Q: What are the limitations of the margin of error?

A: The margin of error has some limitations, including:

• Assumes normal distribution: The margin of error assumes that the population is normally distributed, which may not always be the case.
• Requires large sample size: The margin of error requires a large sample size to be accurate.
• Does not account for bias: The margin of error does not account for bias in the sample.

Q: How do I calculate the margin of error for a proportion?

A: To calculate the margin of error for a proportion, you need to follow these steps:

1. Determine the confidence level: The confidence level is the probability that the sample proportion will be within the margin of error. In this example, we are assuming a 90% confidence level.
2. Calculate the critical value: The critical value is the value from the standard normal distribution that corresponds to the desired confidence level. For a 90% confidence level, the critical value is approximately 1.645.
3. Calculate the standard error: The standard error is the square root of the product of the sample proportion and the sample size divided by the sample size. In this example, the sample proportion is 0.5, and the sample size is 320.
4. Calculate the margin of error: The margin of error is the product of the critical value, the standard error, and the sample proportion.

Q: What is the formula for the margin of error for a proportion?

A: The margin of error formula for a proportion is:

$$\text{Margin of Error} = \text{Critical Value} \times \text{Standard Error} \times \text{Sample Proportion}$$

Where:

• Critical Value: The value from the standard normal distribution that corresponds to the desired confidence level.
• Standard Error: The square root of the product of the sample proportion and the sample size divided by the sample size.
• Sample Proportion: The sample proportion.

Q: Can I use the margin of error to make predictions about the population proportion?

A: Yes, you can use the margin of error to make predictions about the population proportion with a certain level of confidence. However, you need to be aware of the limitations of the margin of error, including:

• Assumes normal distribution: The margin of error assumes that the population is normally distributed, which may not always be the case.
• Requires large sample size: The margin of error requires a large sample size to be accurate.
• Does not account for bias: The margin of error does not account for bias in the sample.