A Sample Of 21 Hummingbirds Was Randomly Selected And Gave A Mean Weight Of 3.13 Grams With A Standard Deviation Of 0.33 Grams. Find The 95% Confidence Interval To Estimate The Population Mean Weight Of Hummingbirds.Degrees Of Freedom (D.F.):

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Introduction

In statistics, a confidence interval is a range of values that is likely to contain the value of an unknown population parameter. It is a way to express the uncertainty in our estimates of a population parameter. In this case, we are interested in estimating the population mean weight of hummingbirds. We have a sample of 21 hummingbirds with a mean weight of 3.13 grams and a standard deviation of 0.33 grams. Our goal is to find the 95% confidence interval for the population mean weight of hummingbirds.

Calculating the Confidence Interval

To calculate the confidence interval, we need to use the following formula:

CI = x̄ ± (Z * (σ / √n))

where:

  • CI is the confidence interval
  • xÌ„ is the sample mean
  • Z is the Z-score corresponding to the desired confidence level
  • σ is the sample standard deviation
  • n is the sample size

In this case, we have:

  • xÌ„ = 3.13 grams
  • σ = 0.33 grams
  • n = 21
  • Z = 1.96 (for a 95% confidence level)

Plugging in these values, we get:

CI = 3.13 ± (1.96 * (0.33 / √21)) CI = 3.13 ± (1.96 * 0.075) CI = 3.13 ± 0.147

Degrees of Freedom (D.F.)

The degrees of freedom (D.F.) is a measure of the amount of information available in a sample. It is calculated as:

D.F. = n - 1

In this case, we have:

  • n = 21
  • D.F. = 21 - 1
  • D.F. = 20

Discussion

The 95% confidence interval for the population mean weight of hummingbirds is (2.983, 3.297) grams. This means that we are 95% confident that the true population mean weight of hummingbirds lies within this range.

The degrees of freedom (D.F.) is 20, which is a relatively small sample size. This means that our confidence interval may not be very accurate. However, it is still a useful estimate of the population mean weight of hummingbirds.

Conclusion

In conclusion, we have found the 95% confidence interval for the population mean weight of hummingbirds to be (2.983, 3.297) grams. This interval is based on a sample of 21 hummingbirds with a mean weight of 3.13 grams and a standard deviation of 0.33 grams. We have also calculated the degrees of freedom (D.F.) to be 20.

References

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

Additional Resources

  • [1] Confidence Interval Calculator: A calculator that can be used to calculate confidence intervals.
  • [2] Degrees of Freedom Calculator: A calculator that can be used to calculate degrees of freedom.

Related Topics

  • [1] Hypothesis Testing: A statistical method used to test a hypothesis about a population parameter.
  • [2] Regression Analysis: A statistical method used to model the relationship between a dependent variable and one or more independent variables.

Keywords

  • Confidence interval
  • Degrees of freedom
  • Population mean
  • Sample mean
  • Standard deviation
  • Z-score
  • Hypothesis testing
  • Regression analysis

Q&A

Q: What is a confidence interval?

A: A confidence interval is a range of values that is likely to contain the value of an unknown population parameter. It is a way to express the uncertainty in our estimates of a population parameter.

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 the true population parameter is likely to lie. This allows us to make inferences about the population based on a sample of data.

Q: How is the confidence interval calculated?

A: The confidence interval is calculated using the following formula:

CI = x̄ ± (Z * (σ / √n))

where:

  • CI is the confidence interval
  • xÌ„ is the sample mean
  • Z is the Z-score corresponding to the desired confidence level
  • σ is the sample standard deviation
  • n is the sample size

Q: What is the Z-score?

A: The Z-score is a measure of how many standard deviations an observation is away from the mean. It is used to determine the confidence level of the interval.

Q: What is the degrees of freedom (D.F.)?

A: The degrees of freedom (D.F.) is a measure of the amount of information available in a sample. It is calculated as:

D.F. = n - 1

Q: How is the degrees of freedom used in the confidence interval calculation?

A: The degrees of freedom is used to determine the critical value of the Z-score, which is used in the confidence interval calculation.

Q: What is the 95% confidence interval for the population mean weight of hummingbirds?

A: The 95% confidence interval for the population mean weight of hummingbirds is (2.983, 3.297) grams.

Q: What does the 95% confidence interval mean?

A: The 95% confidence interval means that we are 95% confident that the true population mean weight of hummingbirds lies within the range of 2.983 to 3.297 grams.

Q: What are some common applications of confidence intervals?

A: Confidence intervals are commonly used in a variety of fields, including medicine, social sciences, and business. They are used to estimate population parameters, such as means and proportions, and to make inferences about the population based on a sample of data.

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

A: Some common mistakes to avoid when calculating confidence intervals include:

  • Using the wrong formula or method
  • Using an incorrect sample size or standard deviation
  • Failing to account for the degrees of freedom
  • Failing to check the assumptions of the method

Q: How can I calculate confidence intervals in practice?

A: To calculate confidence intervals in practice, you can use a variety of methods, including:

  • Using a calculator or computer software
  • Using a statistical package, such as R or Python
  • Using a table or chart to look up the critical value of the Z-score

Conclusion

In conclusion, confidence intervals are a powerful tool for making inferences about a population based on a sample of data. By understanding how to calculate confidence intervals and avoiding common mistakes, you can use this method to make informed decisions in a variety of fields.

References

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

Additional Resources

  • [1] Confidence Interval Calculator: A calculator that can be used to calculate confidence intervals.
  • [2] Degrees of Freedom Calculator: A calculator that can be used to calculate degrees of freedom.

Related Topics

  • [1] Hypothesis Testing: A statistical method used to test a hypothesis about a population parameter.
  • [2] Regression Analysis: A statistical method used to model the relationship between a dependent variable and one or more independent variables.

Keywords

  • Confidence interval
  • Degrees of freedom
  • Population mean
  • Sample mean
  • Standard deviation
  • Z-score
  • Hypothesis testing
  • Regression analysis