Calculating Critical Values

Oct 13, 2025 by magicalceylontours.com 61 views

Introduction

In statistics, a confidence interval is a range of values within which a population parameter is likely to lie. It is a crucial concept in inferential statistics, allowing us to make inferences about a population based on a sample of data. One of the key components of a confidence interval is the critical value, which is used to calculate the margin of error. In this article, we will explore the critical values that correspond to a 95% confidence level and a sample size of 91.

What are Critical Values?

Critical values are the values of a test statistic that separate the region of rejection from the region of non-rejection in a hypothesis test. In the context of confidence intervals, critical values are used to determine the margin of error, which is the maximum amount by which the sample statistic may differ from the population parameter. The critical value is typically denoted by the symbol z or t, depending on whether the population standard deviation is known or unknown.

Calculating Critical Values

To calculate the critical value for a 95% confidence level and a sample size of 91, we need to use the formula for the margin of error:

margin of error = (critical value) * (standard deviation) / sqrt(sample size)

The critical value for a 95% confidence level can be found using a standard normal distribution (Z-distribution) or a t-distribution, depending on whether the population standard deviation is known or unknown.

Using the Z-Distribution

If the population standard deviation is known, we can use the Z-distribution to calculate the critical value. The Z-distribution is a continuous probability distribution that is symmetric about the mean and has a standard deviation of 1. The critical value for a 95% confidence level can be found using the following formula:

critical value = z * sqrt((sample size - 1) / (sample size - 1 + (population standard deviation)^2 / sample variance))

where z is the Z-score corresponding to the desired confidence level.

Using the T-Distribution

If the population standard deviation is unknown, we need to use the t-distribution to calculate the critical value. The t-distribution is a continuous probability distribution that is symmetric about the mean and has a standard deviation that depends on the sample size. The critical value for a 95% confidence level can be found using the following formula:

critical value = t * sqrt((sample size - 1) / (sample size - 1 + (sample variance) / (population standard deviation)^2))

where t is the t-score corresponding to the desired confidence level and sample size.

Calculating the Critical Value for a Sample Size of 91

Using the Z-distribution, we can calculate the critical value for a 95% confidence level and a sample size of 91 as follows:

critical value = 1.96 * sqrt((91 - 1) / (91 - 1 + (population standard deviation)^2 / sample variance))

Assuming a population standard deviation of 10, we get:

critical value = 1.96 * sqrt(90 / 90 + (10^2) / (sample variance))

Simplifying the expression, we get:

critical value = 1.96 * sqrt(1 + (100) / (sample variance))

Using a calculator, we get:

critical value = 1.96 * 1.005 = 1.992

Rounding to three decimal places, we get:

critical value = 1.992

Comparing the Calculated Critical Value with the Given Options

Comparing the calculated critical value with the given options, we see that the closest match is:

A. 65.647, 118.136

However, this option is not the correct answer. The correct answer is not among the given options.

Conclusion

In conclusion, the critical values that correspond to a 95% confidence level and a sample size of 91 are not among the given options. The correct answer requires a more detailed calculation of the critical value using the Z-distribution or t-distribution.

References

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

Note

Q: What is a critical value in statistics?

A: A critical value is a value of a test statistic that separates the region of rejection from the region of non-rejection in a hypothesis test. In the context of confidence intervals, critical values are used to determine the margin of error, which is the maximum amount by which the sample statistic may differ from the population parameter.

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

A: To calculate the critical value for a confidence interval, you need to use the formula for the margin of error:

margin of error = (critical value) * (standard deviation) / sqrt(sample size)

The critical value can be found using a standard normal distribution (Z-distribution) or a t-distribution, depending on whether the population standard deviation is known or unknown.

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

A: A Z-distribution is a continuous probability distribution that is symmetric about the mean and has a standard deviation of 1. It is used when the population standard deviation is known. A t-distribution, on the other hand, is a continuous probability distribution that is symmetric about the mean and has a standard deviation that depends on the sample size. It is used when the population standard deviation is unknown.

Q: How do I choose between a Z-distribution and a t-distribution?

A: You should use a Z-distribution when the population standard deviation is known, and a t-distribution when the population standard deviation is unknown.

Q: What is the formula for the margin of error?

A: The formula for the margin of error is:

margin of error = (critical value) * (standard deviation) / sqrt(sample size)

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 know the critical value, standard deviation, and sample size. You can then plug these values into the formula for the margin of error.

Q: What is the relationship between the confidence level and the critical value?

A: The confidence level and the critical value are inversely related. As the confidence level increases, the critical value decreases.

Q: How do I determine the sample size for a confidence interval?

A: The sample size for a confidence interval depends on the desired margin of error, the population standard deviation, and the confidence level. You can use a sample size calculator or consult with a statistician to determine the required sample size.

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 allows you to make inferences about a population based on a sample of data.

Q: How do I interpret a confidence interval?

A: To interpret a confidence interval, you need to understand the meaning of the interval. The interval represents the range of values within which the population parameter is likely to lie. The width of the interval represents the margin of error, which is the maximum amount by which the sample statistic may differ from the population parameter.

Q: What are some common mistakes to avoid when working with confidence intervals?

A: Some common mistakes to avoid when working with confidence intervals include:

Using the wrong critical value or margin of error
Failing to account for the population standard deviation
Using an incorrect sample size
Interpreting the confidence interval incorrectly

Q: How do I choose the right statistical method for my research question?

A: To choose the right statistical method for your research question, you need to consider the research question, the data, and the desired outcome. You should consult with a statistician or use a statistical software package to determine the best method for your research question.

Q: What are some common statistical methods used in research?

A: Some common statistical methods used in research include:

Hypothesis testing
Confidence intervals
Regression analysis
Analysis of variance (ANOVA)
Non-parametric tests

Q: How do I know if my research question is suitable for a confidence interval?

A: Your research question is suitable for a confidence interval if you want to make inferences about a population parameter based on a sample of data. You should also have a clear understanding of the population parameter and the desired margin of error.