A Simple Random Sample Of Size $n$ Is Drawn From A Normally Distributed Population. The Mean Of The Sample Is $\overline{x}$, And The Standard Deviation Is $s$. What Is The $99\%$ Confidence Interval For The

Introduction

In statistics, a confidence interval is a range of values within which a population parameter is likely to lie. When a simple random sample is drawn from a normally distributed population, we can use the sample mean and standard deviation to construct a confidence interval for the population mean. In this article, we will discuss how to construct a 99% confidence interval for the population mean using the sample mean and standard deviation.

The Formula for the Confidence Interval

The formula for the confidence interval is given by:

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

where:

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

Choosing the Critical Value

To construct a 99% confidence interval, we need to choose the critical value $z_{\alpha/2}$ from the standard normal distribution. The value of $\alpha$ is given by $1 - (1 - \alpha)$, where $\alpha$ is the desired confidence level. In this case, we want a 99% confidence interval, so $\alpha = 0.01$. The critical value $z_{\alpha/2}$ is then given by:

$$z_{\alpha/2} = z_{0.005} = 2.575$$

Calculating the Margin of Error

The margin of error is given by:

$$\frac{s}{\sqrt{n}}$$

This represents the maximum amount by which the sample mean can differ from the population mean.

Constructing the Confidence Interval

To construct the confidence interval, we need to add and subtract the margin of error from the sample mean:

$$\overline{x} \pm 2.575 \frac{s}{\sqrt{n}}$$

This gives us a range of values within which the population mean is likely to lie.

Example

Suppose we have a sample of size $n = 100$ with a sample mean of $\overline{x} = 25$ and a sample standard deviation of $s = 5$. We want to construct a 99% confidence interval for the population mean. Using the formula above, we get:

$$25 \pm 2.575 \frac{5}{\sqrt{100}}$$

$$25 \pm 2.575 \times 0.5$$

$$25 \pm 1.2875$$

This gives us a confidence interval of $(23.7125, 26.2875)$.

Conclusion

In this article, we have discussed how to construct a 99% confidence interval for the population mean using a simple random sample from a normally distributed population. We have used the sample mean and standard deviation to calculate the margin of error and construct the confidence interval. The example illustrates how to apply the formula in practice.

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.

Further Reading

• [1] Confidence Intervals for the Population Mean. Stat Trek.
• [2] Confidence Intervals. Khan Academy.

Tags

• Confidence interval
• Population mean
• Sample mean
• Sample standard deviation
• Critical value
• Margin of error
• Simple random sample
• Normally distributed population
  

Introduction

In our previous article, we discussed how to construct a 99% confidence interval for the population mean using a simple random sample from a normally distributed population. However, we know that there are many questions that readers may have about confidence intervals. In this article, we will address some of the most 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 gives us an idea of the uncertainty associated with a sample estimate.

Q: What is the difference between a confidence interval and a margin of error?

A: A confidence interval is a range of values within which a population parameter is likely to lie, while a margin of error is the maximum amount by which the sample mean can differ from the population mean.

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, and how do I choose it?

A: The critical value is a value from the standard normal distribution that is used to calculate the margin of error. It is chosen based on the desired confidence level.

Q: Can I use a confidence interval with a sample size less than 30?

A: While it is technically possible to use a confidence interval with a sample size less than 30, the results may not be reliable. The sample size should be at least 30 to ensure that the sample is representative of the population.

Q: Can I use a confidence interval with a non-normal population?

A: While the confidence interval formula assumes a normally distributed population, it can still be used with non-normal populations. However, the results may not be reliable, and other methods (e.g. bootstrapping) may be more suitable.

Q: How do I interpret the results of a confidence interval?

A: The results of a confidence interval should be interpreted as follows:

• If the confidence interval includes the population mean, it suggests that the sample mean is a reliable estimate of the population mean.
• If the confidence interval does not include the population mean, it suggests that the sample mean is not a reliable estimate of the population mean.

Q: Can I use a confidence interval to compare two population means?

A: While a confidence interval can be used to compare two population means, it is not the most suitable method. A more suitable method would be to use a hypothesis test (e.g. t-test).

Q: Can I use a confidence interval to estimate a population proportion?

A: While a confidence interval can be used to estimate a population proportion, it is not the most suitable method. A more suitable method would be to use a hypothesis test (e.g. z-test).

Conclusion

In this article, we have addressed some of the most frequently asked questions about confidence intervals. We hope that this article has provided a better understanding of confidence intervals and how to use them in practice.

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.

Further Reading

• [1] Confidence Intervals for the Population Mean. Stat Trek.
• [2] Confidence Intervals. Khan Academy.

Tags

• Confidence interval
• Population mean
• Sample mean
• Sample standard deviation
• Critical value
• Margin of error
• Simple random sample
• Normally distributed population
• Hypothesis test
• Population proportion