Create Confidence Intervals Related To The Interval And Rati ✓ Solved
Create Confidence Intervals Related To The Interval And Ratio Level Da
Create confidence intervals related to the interval and ratio-level data you collected. What is the best estimate of the population mean? Develop a 95% confidence interval for the population mean. Develop a 90% confidence interval for the population mean. Develop a 98% confidence interval for the population mean. Interpret the confidence interval.
Sample Paper For Above instruction
Introduction
Confidence intervals are a fundamental concept in inferential statistics, providing a range within which the true population parameter, such as the mean, is likely to fall with a certain level of confidence. This paper discusses the process of creating confidence intervals based on interval and ratio-level data, focusing on estimating the population mean with 90%, 95%, and 98% confidence levels. The data analyzed are assumed to be representative samples from a population, and the goal is to determine the most accurate estimate of the population mean and interpret the implications of these confidence intervals.
Understanding Confidence Intervals
Confidence intervals (CIs) quantify the uncertainty associated with a sample estimate of a population parameter. In the context of estimating the population mean, the CI provides an interval around the sample mean, within which the true population mean is expected to lie with a specified probability. The level of confidence (e.g., 90%, 95%, 98%) reflects the proportion of such intervals that would contain the true population mean if the same sampling procedure were repeated multiple times.
Data Collection and Descriptive Statistics
For the purposes of this exercise, assume a dataset consisting of interval or ratio-level measurements collected from a sample. For example, these could include measurements such as weights, test scores, or lengths. The sample size (n), sample mean (x̄), and sample standard deviation (s) are computed as the initial descriptive statistics. These values serve as the foundation for constructing confidence intervals.
Calculating Confidence Intervals
The formula used to calculate these confidence intervals depends on whether the population standard deviation (σ) is known.
1. When σ is unknown, as is typically the case, the t-distribution is used:
\[
\text{CI} = \bar{x} \pm t^* \times \left( \frac{s}{\sqrt{n}} \right)
\]
2. \( t^* \) is the critical value from the t-distribution corresponding to the specified confidence level and degrees of freedom (df = n - 1).
For each confidence level (90%, 95%, 98%), the respective critical values are obtained from t-tables or statistical software.
Developing Confidence Intervals
Assuming the sample data provided the following descriptive statistics:
- Sample size (n): 30
- Sample mean (\(\bar{x}\)): 50
- Sample standard deviation (s): 10
Using these, the confidence intervals are calculated as follows:
90% Confidence Interval
- Degrees of freedom: 29
- Critical value (\(t^*\)): approximately 1.699 (from t-tables)
\[
\text{Margin of Error} = 1.699 \times \frac{10}{\sqrt{30}} \approx 3.096
\]
\[
\text{CI} = 50 \pm 3.096 \rightarrow (46.904, 53.096)
\]
95% Confidence Interval
- Critical value (\(t^*\)): approximately 2.045
\[
\text{Margin of Error} = 2.045 \times \frac{10}{\sqrt{30}} \approx 3.733
\]
\[
\text{CI} = 50 \pm 3.733 \rightarrow (46.267, 53.733)
\]
98% Confidence Interval
- Critical value (\(t^*\)): approximately 2.457
\[
\text{Margin of Error} = 2.457 \times \frac{10}{\sqrt{30}} \approx 4.491
\]
\[
\text{CI} = 50 \pm 4.491 \rightarrow (45.509, 54.491)
\]
Interpreting the Confidence Intervals
The constructed confidence intervals provide ranges within which the true population mean is likely to reside, given the confidence level. For example, the 95% confidence interval indicates that there is a 95% probability that the interval (46.267, 53.733) contains the true mean. These intervals account for sample variability and help inform decision-making in research and applied settings.
Conclusion
Creating confidence intervals for the population mean using sample data at multiple confidence levels offers valuable insights into the estimate's precision. As the confidence level increases, the interval widens, reflecting greater certainty but reduced precision. Conversely, narrower intervals at lower confidence levels offer less certainty. Proper interpretation of these intervals is essential for accurate conclusions about the population parameter, especially in fields such as social sciences, healthcare, and economics.
References
Fisher, R. A. (1925). Statistical Methods for Research Workers. Oliver & Boyd.
Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics (8th ed.). Freeman.
Newcombe, R. G. (1998). Two-sided confidence intervals for the single proportion: comparison of seven methods. Statistics in Medicine, 17(8), 857-872.
Cochran, W. G. (1977). Sampling Techniques (3rd ed.). John Wiley & Sons.
Lenth, R. V. (2006). Some Practical Guidelines for Effective Use of Power Analysis. The American Statistician, 60(3), 199-204.