How To Construct A 90 Confidence Interval

How to Construct a 90% Confidence Interval: A Comprehensive Guide

Understanding confidence intervals is crucial for anyone working with statistical data. They provide a range within which a population parameter is likely to lie, giving us a measure of uncertainty surrounding our estimates. This comprehensive guide will walk you through the process of constructing a 90% confidence interval, explaining the underlying concepts and providing practical examples.

What is a Confidence Interval?

A confidence interval is a range of values, calculated from sample data, that is likely to contain the true value of a population parameter with a certain level of confidence. The level of confidence, typically expressed as a percentage (e.g., 90%, 95%, 99%), represents the probability that the interval contains the true population parameter. A 90% confidence interval, therefore, means that if we were to repeat the sampling process many times, 90% of the calculated intervals would contain the true population parameter.

It's crucial to understand that the confidence interval is about the process of estimation, not the specific interval itself. We cannot say with 90% probability that the true value lies within this specific interval. The 90% refers to the long-run frequency of intervals containing the true value.

Key Components of a Confidence Interval

Before diving into the construction process, let's define the essential components:

• Point Estimate: This is the best single guess for the population parameter based on the sample data. For example, the sample mean (x̄) is a point estimate for the population mean (μ).
• Margin of Error: This quantifies the uncertainty in the point estimate. It's the amount added and subtracted from the point estimate to create the interval. A larger margin of error indicates greater uncertainty.
• Confidence Level: This is the probability that the interval contains the true population parameter. A 90% confidence level implies a 10% chance that the interval does not contain the true value.
• Critical Value: This value, denoted by z* or t*, depends on the confidence level and the distribution of the sample statistic (e.g., z-distribution for large samples, t-distribution for small samples). It determines the width of the confidence interval.

Constructing a 90% Confidence Interval for a Population Mean

The method for constructing a confidence interval varies slightly depending on whether the population standard deviation (σ) is known or unknown.

Case 1: Population Standard Deviation (σ) is Known

When the population standard deviation is known, we use the z-distribution to calculate the confidence interval. The formula is:

Confidence Interval = Point Estimate ± Margin of Error

Where:

• Point Estimate = x̄ (sample mean)

• Margin of Error = z * (σ / √n)*

  • z* is the critical value corresponding to the 90% confidence level. For a 90% confidence level, the alpha (α) level is 1 - 0.90 = 0.10. Since we use a two-tailed test, we divide alpha by 2 (0.10/2 = 0.05). Looking up the z-score corresponding to 0.05 in a z-table or using statistical software gives us z* ≈ 1.645.
  • σ is the population standard deviation.
  • n is the sample size.

Example: Suppose we have a sample of n = 100 observations with a sample mean (x̄) of 50 and a known population standard deviation (σ) of 10. To construct a 90% confidence interval:

1. Find the critical value (z):* z* ≈ 1.645 (as explained above).
2. Calculate the margin of error: Margin of Error = 1.645 * (10 / √100) = 1.645
3. Construct the confidence interval: Confidence Interval = 50 ± 1.645 = (48.355, 51.645)

Therefore, we are 90% confident that the true population mean lies between 48.355 and 51.645.

Case 2: Population Standard Deviation (σ) is Unknown

When the population standard deviation is unknown, which is more common in real-world scenarios, we estimate it using the sample standard deviation (s). In this case, we use the t-distribution instead of the z-distribution. The formula becomes:

Confidence Interval = Point Estimate ± Margin of Error

Where:

• Point Estimate = x̄ (sample mean)

• Margin of Error = t * (s / √n)*

  • t* is the critical value from the t-distribution with n-1 degrees of freedom and corresponding to the 90% confidence level. The degrees of freedom (df) equals the sample size (n) minus 1.
  • s is the sample standard deviation.
  • n is the sample size.

Example: Let's say we have a sample of n = 25 observations with a sample mean (x̄) of 75 and a sample standard deviation (s) of 5. To construct a 90% confidence interval:

1. Determine the degrees of freedom: df = 25 - 1 = 24
2. Find the critical value (t):* Using a t-table or statistical software with df = 24 and a 90% confidence level (α = 0.10, two-tailed), we find t* ≈ 1.711.
3. Calculate the margin of error: Margin of Error = 1.711 * (5 / √25) = 1.711
4. Construct the confidence interval: Confidence Interval = 75 ± 1.711 = (73.289, 76.711)

Therefore, we are 90% confident that the true population mean lies between 73.289 and 76.711.

Factors Affecting the Width of the Confidence Interval

The width of the confidence interval reflects the precision of our estimate. A narrower interval indicates a more precise estimate, while a wider interval suggests greater uncertainty. Several factors influence the width:

• Sample Size (n): Larger sample sizes lead to narrower confidence intervals because they provide more information about the population.
• Confidence Level: Higher confidence levels (e.g., 99% instead of 90%) result in wider intervals because they require a larger margin of error to ensure greater certainty.
• Population Standard Deviation (σ or s): Larger standard deviations lead to wider intervals, indicating greater variability in the data.

Interpreting Confidence Intervals

It's vital to correctly interpret a confidence interval. The following points are crucial:

• It's about the process: The 90% refers to the reliability of the method, not the specific interval itself.
• Not a probability statement about the population parameter: We don't say there's a 90% probability the true value is within the interval.
• It reflects uncertainty: A wider interval reflects greater uncertainty about the true population parameter.
• It’s dependent on sample data: Different samples will yield different confidence intervals.

Confidence Intervals for Other Parameters

The principles discussed above can be extended to construct confidence intervals for other population parameters, such as:

• Population Proportion: Used when dealing with categorical data (e.g., the proportion of people who support a particular candidate).
• Difference between Two Means: Used to compare the means of two populations.
• Difference between Two Proportions: Used to compare the proportions of two populations.

The formulas and critical values will differ depending on the specific parameter and the assumptions made about the data.

Using Software for Confidence Interval Calculations

Statistical software packages like R, SPSS, SAS, and Python (with libraries like SciPy) can easily calculate confidence intervals. These tools automate the calculations and provide accurate results, saving time and reducing the risk of errors.

Conclusion

Constructing a 90% confidence interval is a fundamental statistical procedure that provides a valuable range for estimating a population parameter. Understanding the underlying principles, formulas, and interpretations is essential for anyone working with data analysis. Remember to choose the appropriate method based on whether the population standard deviation is known or unknown, and always consider the factors that influence the width of the interval. Using statistical software can streamline the process and enhance accuracy. By mastering these techniques, you'll gain valuable insights from your data and make more informed decisions.