Determining the Degrees of Freedom

Oct 13, 2025 by windailys.com 104 views

Introduction

When constructing a confidence interval for the population mean based on a random sample, it is essential to determine the critical value of the t-distribution. This value is used to calculate the margin of error and ensure that the confidence interval is accurate. In this article, we will discuss how to find the critical value of t for a 95% confidence interval based on a random sample of 15 observations.

Understanding the t-Distribution

The t-distribution is a probability distribution that is used to calculate the critical value for a confidence interval when the sample size is small. It is a continuous probability distribution that is similar to the standard normal distribution but has a slightly different shape. The t-distribution is used when the population standard deviation is unknown, and the sample size is small.

Determining the Degrees of Freedom

To find the critical value of t, we need to determine the degrees of freedom. The degrees of freedom is the number of observations in the sample minus one. In this case, we have a random sample of 15 observations, so the degrees of freedom is 15 - 1 = 14.

Finding the Critical Value of t

To find the critical value of t, we need to use a t-distribution table or a calculator. The t-distribution table provides the critical values of t for different degrees of freedom and confidence levels. We are looking for the critical value of t for a 95% confidence interval with 14 degrees of freedom.

Using a T-Distribution Table

A t-distribution table is a table that provides the critical values of t for different degrees of freedom and confidence levels. The table is typically arranged in a grid format, with the degrees of freedom on one axis and the confidence level on the other axis. We can use the table to find the critical value of t for a 95% confidence interval with 14 degrees of freedom.

Using a Calculator

Alternatively, we can use a calculator to find the critical value of t. Most calculators have a built-in function for calculating the critical value of t. We can enter the degrees of freedom and the confidence level, and the calculator will provide the critical value of t.

Conclusion

In conclusion, to find the critical value of t for a 95% confidence interval based on a random sample of 15 observations, we need to determine the degrees of freedom and use a t-distribution table or a calculator to find the critical value of t. The critical value of t is essential for calculating the margin of error and ensuring that the confidence interval is accurate.

Example

Suppose we have a random sample of 15 observations, and we want to construct a 95% confidence interval for the population mean. We can use the critical value of t to calculate the margin of error and construct the confidence interval.

Step 1: Determine the Degrees of Freedom

The degrees of freedom is the number of observations in the sample minus one. In this case, we have a random sample of 15 observations, so the degrees of freedom is 15 - 1 = 14.

Step 2: Find the Critical Value of t

We can use a t-distribution table or a calculator to find the critical value of t for a 95% confidence interval with 14 degrees of freedom. The critical value of t is approximately 2.145.

Step 3: Calculate the Margin of Error

The margin of error is calculated by multiplying the critical value of t by the standard error of the mean. The standard error of the mean is calculated by dividing the sample standard deviation by the square root of the sample size.

Step 4: Construct the Confidence Interval

The confidence interval is constructed by adding and subtracting the margin of error from the sample mean.

Final Answer

The final answer is: $\boxed{2.145}$

Introduction

In our previous article, we discussed how to find the critical value of t for a 95% confidence interval based on a random sample of 15 observations. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

Q: What is the t-distribution and why is it used?

A: The t-distribution is a probability distribution that is used to calculate the critical value for a confidence interval when the sample size is small. It is a continuous probability distribution that is similar to the standard normal distribution but has a slightly different shape. The t-distribution is used when the population standard deviation is unknown, and the sample size is small.

Q: How do I determine the degrees of freedom?

A: The degrees of freedom is the number of observations in the sample minus one. In this case, we have a random sample of 15 observations, so the degrees of freedom is 15 - 1 = 14.

Q: How do I find the critical value of t?

A: You can use a t-distribution table or a calculator to find the critical value of t. The t-distribution table provides the critical values of t for different degrees of freedom and confidence levels. Alternatively, you can use a calculator to find the critical value of t.

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

A: A t-distribution table is a table that provides the critical values of t for different degrees of freedom and confidence levels. A calculator, on the other hand, can be used to find the critical value of t by entering the degrees of freedom and the confidence level.

Q: How do I calculate the margin of error?

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

Q: How do I construct the confidence interval?

A: The confidence interval is constructed by adding and subtracting the margin of error from the sample mean.

Q: What is the final answer?

A: The final answer is the critical value of t, which is approximately 2.145.

Q: Can I use a different confidence level?

A: Yes, you can use a different confidence level. However, you will need to find the corresponding critical value of t for the new confidence level.

Q: Can I use a different sample size?

A: Yes, you can use a different sample size. However, you will need to find the corresponding degrees of freedom and critical value of t for the new sample size.

Q: What if I have a large sample size?

A: If you have a large sample size, you can use the standard normal distribution instead of the t-distribution.

Q: What if I have a small sample size and the population standard deviation is known?

A: If you have a small sample size and the population standard deviation is known, you can use the standard normal distribution instead of the t-distribution.

Final Answer

The final answer is: $\boxed{2.145}$