Give A $99.8%$ Confidence Interval For Μ 1 − Μ 2 \mu_1 - \mu_2 Μ 1 ​ − Μ 2 ​ Given The Following Information: N 1 = 30 , X ˉ 1 = 2.28 , S 1 = 0.96 N 2 = 55 , X ˉ 2 = 2.73 , S 2 = 0.65 \begin{array}{l} n_1 = 30, \bar{x}_1 = 2.28, S_1 = 0.96 \\ n_2 = 55, \bar{x}_2 = 2.73, S_2 = 0.65 \end{array} N 1 ​ = 30 , X ˉ 1 ​ = 2.28 , S 1 ​ = 0.96 N 2 ​ = 55 , X ˉ 2 ​ = 2.73 , S 2 ​ = 0.65 ​ $\square \pm

Introduction

In statistics, it is often necessary to compare the means of two populations to determine if there is a significant difference between them. One way to do this is by calculating the confidence interval for the difference between the two population means. In this article, we will provide a step-by-step guide on how to calculate a 99.8% confidence interval for the difference between two population means, given the sample means, sample standard deviations, and sample sizes.

Confidence Interval Formula

The formula for the confidence interval for the difference between two population means is:

$$\left(\bar{x}_1 - \bar{x}_2\right) \pm t_{\alpha/2} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$$

where:

• $\bar{x}_1$ and $\bar{x}_2$ are the sample means
• $s_1$ and $s_2$ are the sample standard deviations
• $n_1$ and $n_2$ are the sample sizes
• $t_{\alpha/2}$ is the critical value from the t-distribution with $n_1 + n_2 - 2$ degrees of freedom

Given Information

We are given the following information:

• $n_1 = 30$
• $\bar{x}_1 = 2.28$
• $s_1 = 0.96$
• $n_2 = 55$
• $\bar{x}_2 = 2.73$
• $s_2 = 0.65$

Step 1: Calculate the Standard Error

The standard error is calculated as:

$$\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$$

Plugging in the values, we get:

$$\sqrt{\frac{(0.96)^2}{30} + \frac{(0.65)^2}{55}} = \sqrt{\frac{0.9216}{30} + \frac{0.4225}{55}} = \sqrt{0.03072 + 0.00769} = \sqrt{0.03841} = 0.196$$

Step 2: Determine the Critical Value

The critical value is determined using the t-distribution with $n_1 + n_2 - 2$ degrees of freedom. In this case, we have:

$$n_1 + n_2 - 2 = 30 + 55 - 2 = 83$$

Using a t-distribution table or calculator, we find that the critical value for a 99.8% confidence interval with 83 degrees of freedom is approximately 2.99.

Step 3: Calculate the Confidence Interval

Now that we have the standard error and critical value, we can calculate the confidence interval:

$$\left(\bar{x}_1 - \bar{x}_2\right) \pm t_{\alpha/2} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$$

Plugging in the values, we get:

$$\left(2.28 - 2.73\right) \pm 2.99 \times 0.196 = -0.45 \pm 0.58$$

Conclusion

In this article, we provided a step-by-step guide on how to calculate a 99.8% confidence interval for the difference between two population means. We used the given information to calculate the standard error, critical value, and confidence interval. The resulting confidence interval is:

$$-0.45 \pm 0.58$$

This means that we are 99.8% confident that the true difference between the two population means lies within the interval $-1.03$ to $0.13$.

Discussion

The confidence interval provides a range of values within which the true difference between the two population means is likely to lie. In this case, the interval is quite wide, indicating that there is a significant amount of uncertainty in our estimate of the difference between the two population means.

One possible explanation for the wide interval is that the sample sizes are relatively small, which can lead to a loss of precision in our estimates. Additionally, the sample standard deviations are quite different, which can also contribute to the uncertainty in our estimates.

In conclusion, the confidence interval provides a useful tool for understanding the uncertainty in our estimates of the difference between two population means. By considering the width of the interval and the potential sources of uncertainty, we can gain a better understanding of the reliability of our estimates.

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. (2018). Elementary statistics: Picturing the world. Cengage Learning.

Appendix

The following is a summary of the calculations performed in this article:

Step	Calculation
1	$\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = 0.196$
2	$t_{\alpha/2} = 2.99$
3	$\left(\bar{x}_1 - \bar{x}_2\right) \pm t_{\alpha/2} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = -0.45 \pm 0.58$

Introduction

In our previous article, we provided a step-by-step guide on how to calculate a 99.8% confidence interval for the difference between two population means. In this article, we will answer some frequently asked questions related to confidence intervals and provide additional insights into the topic.

Q: What is a confidence interval?

A confidence interval is a range of values within which a population parameter is likely to lie. It is calculated using a sample of data and provides a measure of the uncertainty in our estimates.

Q: Why do we need confidence intervals?

Confidence intervals are useful because they provide a range of values within which the true population parameter is likely to lie. This allows us to make informed decisions and take action based on our estimates.

Q: How do I choose the level of confidence?

The level of confidence is typically chosen before collecting data. Common levels of confidence include 90%, 95%, and 99%. A higher level of confidence requires a larger sample size and provides a narrower interval.

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

A confidence interval is a range of values within which the true population parameter is likely to lie. A margin of error is the maximum amount by which the estimate may differ from the true value.

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

Yes, you can use a confidence interval to compare two population means. However, you need to make sure that the samples are independent and that the data meet the assumptions of the test.

Q: What are the assumptions of the test?

The assumptions of the test include:

• The data are normally distributed
• The samples are independent
• The population variances are equal

Q: How do I check the assumptions of the test?

You can check the assumptions of the test by:

• Plotting the data to check for normality
• Calculating the correlation between the samples
• Performing a test for equal variances

Q: What if my data do not meet the assumptions of the test?

If your data do not meet the assumptions of the test, you may need to transform the data or use a different test.

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

Yes, you can use a confidence interval to compare more than two population means. However, you need to make sure that the samples are independent and that the data meet the assumptions of the test.

Q: What are some common applications of confidence intervals?

Confidence intervals are commonly used in:

• Quality control
• Public health
• Business
• Social sciences

Conclusion

In this article, we answered some frequently asked questions related to confidence intervals and provided additional insights into the topic. We hope that this article has been helpful in understanding the concept of confidence intervals and how to apply 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. (2018). Elementary statistics: Picturing the world. Cengage Learning.

Appendix

The following is a summary of the questions and answers in this article:

Q	A
What is a confidence interval?	A confidence interval is a range of values within which a population parameter is likely to lie.
Why do we need confidence intervals?	Confidence intervals are useful because they provide a range of values within which the true population parameter is likely to lie.
How do I choose the level of confidence?	The level of confidence is typically chosen before collecting data. Common levels of confidence include 90%, 95%, and 99%.
What is the difference between a confidence interval and a margin of error?	A confidence interval is a range of values within which the true population parameter is likely to lie. A margin of error is the maximum amount by which the estimate may differ from the true value.
Can I use a confidence interval to compare two population means?	Yes, you can use a confidence interval to compare two population means. However, you need to make sure that the samples are independent and that the data meet the assumptions of the test.
What are the assumptions of the test?	The assumptions of the test include: the data are normally distributed, the samples are independent, and the population variances are equal.
How do I check the assumptions of the test?	You can check the assumptions of the test by plotting the data to check for normality, calculating the correlation between the samples, and performing a test for equal variances.
What if my data do not meet the assumptions of the test?	If your data do not meet the assumptions of the test, you may need to transform the data or use a different test.
Can I use a confidence interval to compare more than two population means?	Yes, you can use a confidence interval to compare more than two population means. However, you need to make sure that the samples are independent and that the data meet the assumptions of the test.
What are some common applications of confidence intervals?	Confidence intervals are commonly used in quality control, public health, business, and social sciences.