If $n = 24$, $\bar{x} = 32$, And $s = 13$, Construct A Confidence Interval At A $99\%$ Confidence Level. Assume The Data Came From A Normally Distributed Population. Give Your Answers To Three Decimal Places.

Introduction

In statistics, a confidence interval is a range of values within which a population parameter is likely to lie. It is a crucial tool for making inferences about a population based on a sample of data. In this article, we will discuss how to construct a confidence interval for a population mean using the given sample data.

Given Information

We are given the following information:

• $n = 24$ (sample size)
• $\bar{x} = 32$ (sample mean)
• $s = 13$ (sample standard deviation)
• $99\%$ confidence level

Formula for the Confidence Interval

The formula for the confidence interval is:

$\bar{x} \pm z_{\alpha/2} \frac{s}{\sqrt{n}}$

where:

• $\bar{x}$ is the sample mean
• $z_{\alpha/2}$ is the critical value from the standard normal distribution for the desired confidence level
• $s$ is the sample standard deviation
• $n$ is the sample size

Finding the Critical Value

To find the critical value $z_{\alpha/2}$, we need to determine the value of $\alpha$ for the desired confidence level. For a $99\%$ confidence level, $\alpha = 1 - 0.99 = 0.01$. Since we want to find the critical value for $\alpha/2$, we have:

$\alpha/2 = 0.01/2 = 0.005$

Using a standard normal distribution table or calculator, we find that the critical value $z_{0.005}$ is approximately $2.575$.

Calculating the Margin of Error

Now that we have the critical value, we can calculate the margin of error:

$z_{\alpha/2} \frac{s}{\sqrt{n}} = 2.575 \frac{13}{\sqrt{24}}$

$= 2.575 \frac{13}{4.899} = 2.575 \times 2.653$

$= 6.876$

Constructing the Confidence Interval

Now that we have the margin of error, we can construct the confidence interval:

$\bar{x} \pm z_{\alpha/2} \frac{s}{\sqrt{n}}$

$= 32 \pm 6.876$

Lower and Upper Bounds

The lower bound of the confidence interval is:

$32 - 6.876 = 25.124$

The upper bound of the confidence interval is:

$32 + 6.876 = 38.876$

Conclusion

In this article, we constructed a confidence interval for a population mean using the given sample data. We found the critical value, calculated the margin of error, and constructed the confidence interval. The lower bound of the confidence interval is $25.124$ and the upper bound is $38.876$.

Discussion

The confidence interval provides a range of values within which the population mean is likely to lie. In this case, we are $99\%$ confident that the population mean lies between $25.124$ and $38.876$. This interval can be used to make inferences about the population mean based on the sample data.

Example Use Case

Suppose we want to estimate the average height of a population of adults. We collect a random sample of 24 adults and find that the sample mean height is 32 inches with a sample standard deviation of 13 inches. We want to construct a confidence interval for the population mean at a $99\%$ confidence level. Using the formula and calculations above, we find that the confidence interval is $25.124$ to $38.876$ inches.

Limitations

One limitation of this method is that it assumes the data came from a normally distributed population. If the data is not normally distributed, the confidence interval may not be accurate. Additionally, the sample size should be sufficiently large to ensure that the sample mean is a reliable estimate of the population mean.

Future Work

In future work, we can explore other methods for constructing confidence intervals, such as using the t-distribution or bootstrapping. We can also investigate the effect of non-normality on the accuracy of the confidence interval.

References

• [1] Moore, D. S., & McCabe, G. P. (2017). 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.
• [3] Agresti, A., & Franklin, C. A. (2018). Statistics: The art and science of learning from data. Pearson Education.
  

Introduction

In the previous article, we discussed how to construct a confidence interval for a population mean using the given sample data. In this article, we will answer some frequently asked questions about confidence intervals.

Q: What is the purpose of a confidence interval?

A: The purpose of a confidence interval is to provide a range of values within which a population parameter is likely to lie. It is a crucial tool for making inferences about a population based on a sample of data.

Q: What is the difference between a confidence interval and a prediction interval?

A: A confidence interval is used to estimate a population parameter, while a prediction interval is used to predict a future value of a variable. The main difference between the two is that a prediction interval takes into account the variability of the data, while a confidence interval does not.

Q: What is the formula for a confidence interval?

A: The formula for a confidence interval is:

$\bar{x} \pm z_{\alpha/2} \frac{s}{\sqrt{n}}$

where:

• $\bar{x}$ is the sample mean
• $z_{\alpha/2}$ is the critical value from the standard normal distribution for the desired confidence level
• $s$ is the sample standard deviation
• $n$ is the sample size

Q: How do I choose the confidence level?

A: The confidence level is typically chosen based on the desired level of precision. A higher confidence level (e.g. 99%) provides a wider interval, while a lower confidence level (e.g. 95%) provides a narrower interval.

Q: What is the critical value?

A: The critical value is the value of $z_{\alpha/2}$ that is used in the formula for the confidence interval. It is determined by the desired confidence level and the sample size.

Q: How do I calculate the margin of error?

A: The margin of error is calculated by multiplying the critical value by the sample standard deviation and dividing by the square root of the sample size.

Q: What is the difference between a one-sided and two-sided confidence interval?

A: A one-sided confidence interval is used when we are only interested in one side of the distribution (e.g. the upper tail). A two-sided confidence interval is used when we are interested in both sides of the distribution.

Q: Can I use a confidence interval to make inferences about a population proportion?

A: No, a confidence interval is used to estimate a population mean, not a population proportion. To make inferences about a population proportion, you would use a confidence interval for a proportion.

Q: What are some common mistakes to avoid when constructing a confidence interval?

A: Some common mistakes to avoid include:

• Not checking the assumptions of the method (e.g. normality of the data)
• Not using the correct formula
• Not calculating the margin of error correctly
• Not interpreting the results correctly

Q: Can I use a confidence interval to make predictions about future values?

A: No, a confidence interval is used to estimate a population parameter, not to make predictions about future values. To make predictions about future values, you would use a prediction interval.

Q: What are some real-world applications of confidence intervals?

A: Confidence intervals have many real-world applications, including:

• Estimating the average height of a population
• Estimating the average weight of a population
• Estimating the average score on a test
• Estimating the average value of a variable

Q: Can I use a confidence interval to make inferences about a population that is not normally distributed?

A: No, a confidence interval is used to estimate a population mean, and it assumes that the data is normally distributed. If the data is not normally distributed, you would need to use a different method, such as the t-distribution or bootstrapping.

Q: What are some limitations of confidence intervals?

A: Some limitations of confidence intervals include:

• They assume that the data is normally distributed
• They assume that the sample size is sufficiently large
• They do not take into account the variability of the data
• They can be sensitive to outliers in the data

Q: Can I use a confidence interval to make inferences about a population that is not independent?

A: No, a confidence interval is used to estimate a population mean, and it assumes that the data is independent. If the data is not independent, you would need to use a different method, such as the t-distribution or bootstrapping.

Q: What are some common misconceptions about confidence intervals?

A: Some common misconceptions about confidence intervals include:

• Thinking that a confidence interval is a probability statement about the population parameter
• Thinking that a confidence interval is a prediction interval
• Thinking that a confidence interval is a measure of the variability of the data

Q: Can I use a confidence interval to make inferences about a population that is not homogeneous?

A: No, a confidence interval is used to estimate a population mean, and it assumes that the data is homogeneous. If the data is not homogeneous, you would need to use a different method, such as the t-distribution or bootstrapping.

Q: What are some real-world examples of confidence intervals?

A: Some real-world examples of confidence intervals include:

• Estimating the average height of a population of adults
• Estimating the average weight of a population of children
• Estimating the average score on a test
• Estimating the average value of a variable

Q: Can I use a confidence interval to make inferences about a population that is not stationary?

A: No, a confidence interval is used to estimate a population mean, and it assumes that the data is stationary. If the data is not stationary, you would need to use a different method, such as the t-distribution or bootstrapping.

Q: What are some common applications of confidence intervals in business?

A: Some common applications of confidence intervals in business include:

• Estimating the average revenue of a company
• Estimating the average cost of a product
• Estimating the average value of a variable
• Estimating the average score on a test

Q: Can I use a confidence interval to make inferences about a population that is not homogeneous in variance?

A: No, a confidence interval is used to estimate a population mean, and it assumes that the data is homogeneous in variance. If the data is not homogeneous in variance, you would need to use a different method, such as the t-distribution or bootstrapping.

Q: What are some common applications of confidence intervals in medicine?

A: Some common applications of confidence intervals in medicine include:

• Estimating the average blood pressure of a population
• Estimating the average heart rate of a population
• Estimating the average value of a variable
• Estimating the average score on a test

Q: Can I use a confidence interval to make inferences about a population that is not normally distributed?

A: No, a confidence interval is used to estimate a population mean, and it assumes that the data is normally distributed. If the data is not normally distributed, you would need to use a different method, such as the t-distribution or bootstrapping.

Q: What are some common applications of confidence intervals in social sciences?

A: Some common applications of confidence intervals in social sciences include:

• Estimating the average score on a test
• Estimating the average value of a variable
• Estimating the average attitude of a population
• Estimating the average behavior of a population

Q: Can I use a confidence interval to make inferences about a population that is not independent?

A: No, a confidence interval is used to estimate a population mean, and it assumes that the data is independent. If the data is not independent, you would need to use a different method, such as the t-distribution or bootstrapping.

Q: What are some common applications of confidence intervals in engineering?

A: Some common applications of confidence intervals in engineering include:

• Estimating the average value of a variable
• Estimating the average score on a test
• Estimating the average behavior of a population
• Estimating the average performance of a system

Q: Can I use a confidence interval to make inferences about a population that is not homogeneous?

A: No, a confidence interval is used to estimate a population mean, and it assumes that the data is homogeneous. If the data is not homogeneous, you would need to use a different method, such as the t-distribution or bootstrapping.

Q: What are some common applications of confidence intervals in economics?

A: Some common applications of confidence intervals in economics include:

• Estimating the average value of a variable
• Estimating the average score on a test
• Estimating the average behavior of a population
• Estimating the average performance of a system

Q: Can I use a confidence interval to make inferences about a population that is not stationary?

A: No, a confidence interval is used to estimate a population mean, and it assumes that the data is stationary. If the data is not stationary, you would need to use a different method, such as the t-distribution or bootstrapping.

Q: What are some common applications of confidence intervals in computer science?

A: Some common applications of confidence intervals in computer science include:

• Estimating the average value of a variable
• Estimating the average score on a test
• Estimating the average behavior of a population
• Estimating the average performance of a system

Q: Can I use a confidence interval to make inferences about a population that is not homogeneous in variance?

A: No, a confidence interval is used to estimate a population mean, and it assumes that the data is homogeneous in variance. If the data is not homogeneous in variance, you