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What is the Critical Value $t$ for Constructing a $99\%$ Confidence Interval for a Mean from a Sample Size of $n=10$ Observations?

Understanding the Concept of Critical Value

In statistics, a critical value is a value on the standard normal distribution (Z-distribution) or a t-distribution that separates the region of rejection from the region of non-rejection in a hypothesis test. It is a threshold value that determines whether a null hypothesis should be rejected or not. In the context of constructing a confidence interval, the critical value is used to determine the margin of error, which is the maximum amount by which the sample mean is expected to differ from the population mean.

The Role of Sample Size in Determining the Critical Value

The sample size is an important factor in determining the critical value for constructing a confidence interval. A larger sample size generally results in a smaller margin of error, which means that the confidence interval will be narrower. Conversely, a smaller sample size results in a larger margin of error, which means that the confidence interval will be wider.

The T-Distribution and Its Relationship to the Critical Value

The t-distribution is a probability distribution that is used to calculate the critical value for constructing a confidence interval when the sample size is small (typically less than 30). The t-distribution is similar to the standard normal distribution (Z-distribution), but it has a slightly different shape and is more sensitive to outliers. The critical value for the t-distribution is determined by the degrees of freedom, which is equal to the sample size minus one.

Calculating the Critical Value for a 99% Confidence Interval

To calculate the critical value for a 99% confidence interval, we need to use a t-distribution table or calculator. The critical value will depend on the sample size, the desired level of confidence, and the degrees of freedom. In this case, we are given a sample size of $n=10$ observations, and we want to construct a 99% confidence interval.

Using a T-Distribution Table or Calculator

Using a t-distribution table or calculator, we can find the critical value for a 99% confidence interval with a sample size of $n=10$ observations. The critical value is typically denoted as $t_{\alpha/2}$, where $\alpha$ is the desired level of significance (1 - 0.99 = 0.01) and $n-1$ is the degrees of freedom.

Finding the Critical Value

After consulting a t-distribution table or calculator, we find that the critical value for a 99% confidence interval with a sample size of $n=10$ observations is:

$t_{0.005} = 3.169$

This means that the critical value for constructing a 99% confidence interval from a sample size of $n=10$ observations is $t=3.169$.

Conclusion

In conclusion, the critical value $t$ for constructing a 99% confidence interval for a mean from a sample size of $n=10$ observations is $t=3.169$. This value is determined by the t-distribution and is used to calculate the margin of error, which is the maximum amount by which the sample mean is expected to differ from the population mean.

References

• Moore, D. S., & McCabe, G. P. (2011). Introduction to the practice of statistics. W.H. Freeman and Company.
• Larson, R. E., & Farber, B. A. (2013). Elementary statistics: Picturing the world. Cengage Learning.

What is the Critical Value $t$ for Constructing a 99% Confidence Interval for a Mean from a Sample Size of $n=10$ Observations?

The critical value $t$ for constructing a 99% confidence interval for a mean from a sample size of $n=10$ observations is $t=3.169$. This value is determined by the t-distribution and is used to calculate the margin of error, which is the maximum amount by which the sample mean is expected to differ from the population mean.

Answer

A. $t=3.169$
Frequently Asked Questions (FAQs) About Critical Values and Confidence Intervals

Q: What is a critical value, and why is it important in statistics?

A: A critical value is a value on the standard normal distribution (Z-distribution) or a t-distribution that separates the region of rejection from the region of non-rejection in a hypothesis test. It is a threshold value that determines whether a null hypothesis should be rejected or not. Critical values are important in statistics because they help us determine the margin of error, which is the maximum amount by which the sample mean is expected to differ from the population mean.

Q: What is the difference between a Z-distribution and a t-distribution?

A: The Z-distribution and the t-distribution are both probability distributions used in statistics, but they have some key differences. The Z-distribution is used when the sample size is large (typically greater than 30), while the t-distribution is used when the sample size is small (typically less than 30). The t-distribution is more sensitive to outliers and has a slightly different shape than the Z-distribution.

Q: How do I determine the critical value for a confidence interval?

A: To determine the critical value for a confidence interval, you need to use a t-distribution table or calculator. The critical value will depend on the sample size, the desired level of confidence, and the degrees of freedom. The degrees of freedom is equal to the sample size minus one.

Q: What is the relationship between the sample size and the critical value?

A: The sample size is an important factor in determining the critical value for constructing a confidence interval. A larger sample size generally results in a smaller margin of error, which means that the confidence interval will be narrower. Conversely, a smaller sample size results in a larger margin of error, which means that the confidence interval will be wider.

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

A: To calculate the margin of error for a confidence interval, you need to use the critical value and the sample standard deviation. The margin of error is calculated as follows:

Margin of Error = (Critical Value x Sample Standard Deviation) / sqrt(n)

Where n is the sample size.

Q: What is the difference between a 95% confidence interval and a 99% confidence interval?

A: A 95% confidence interval and a 99% confidence interval are both used to estimate a population parameter, but they have different levels of confidence. A 95% confidence interval has a 95% level of confidence, while a 99% confidence interval has a 99% level of confidence. This means that the 99% confidence interval is wider than the 95% confidence interval.

Q: How do I choose the correct critical value for my confidence interval?

A: To choose the correct critical value for your confidence interval, you need to consider the following factors:

• The desired level of confidence
• The sample size
• The degrees of freedom

You can use a t-distribution table or calculator to find the correct critical value.

Q: What is the importance of critical values in hypothesis testing?

A: Critical values are important in hypothesis testing because they help us determine whether to reject or fail to reject the null hypothesis. If the test statistic is greater than the critical value, we reject the null hypothesis. If the test statistic is less than the critical value, we fail to reject the null hypothesis.

Q: Can I use a Z-distribution instead of a t-distribution for small sample sizes?

A: No, you should not use a Z-distribution for small sample sizes. The Z-distribution is used when the sample size is large (typically greater than 30), while the t-distribution is used when the sample size is small (typically less than 30). Using a Z-distribution for small sample sizes can lead to incorrect conclusions.

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

A: To interpret the results of a confidence interval, you need to consider the following factors:

• The confidence level
• The sample size
• The margin of error

A confidence interval provides a range of values within which the population parameter is likely to lie. The confidence level indicates the probability that the population parameter lies within the interval.

Q: Can I use a confidence interval to test a hypothesis?

A: Yes, you can use a confidence interval to test a hypothesis. A confidence interval provides a range of values within which the population parameter is likely to lie. If the null hypothesis is true, the confidence interval should contain the population parameter. If the null hypothesis is false, the confidence interval should not contain the population parameter.

Q: What is the relationship between confidence intervals and hypothesis testing?

A: Confidence intervals and hypothesis testing are related concepts in statistics. A confidence interval provides a range of values within which the population parameter is likely to lie, while hypothesis testing involves testing a null hypothesis against an alternative hypothesis. Confidence intervals can be used to test hypotheses by checking whether the null hypothesis is contained within the interval.