How To Find A 90 Confidence Interval

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

Determining the confidence interval is a crucial aspect of statistical inference. It allows us to estimate a population parameter (like the mean or proportion) with a certain level of confidence, acknowledging the inherent uncertainty in using sample data to make inferences about a larger population. This comprehensive guide will delve into the methods of calculating a 90% confidence interval, exploring different scenarios and providing practical examples.

Understanding Confidence Intervals

Before diving into the calculations, let's establish a foundational understanding. A confidence interval provides a range of values within which we are confident (at a specified level, in this case, 90%) that the true population parameter lies. It doesn't mean there's a 90% chance the true value is within that specific interval; instead, it means that if we were to repeat the sampling process many times, 90% of the calculated intervals would contain the true population parameter.

The key components of a confidence interval are:

• Point Estimate: This is the best single guess of the population parameter based on the sample data. For example, the sample mean is a point estimate of the population mean.
• Margin of Error: This quantifies the uncertainty associated with the point estimate. It represents the amount added and subtracted from the point estimate to create the interval.
• Confidence Level: This is the probability (expressed as a percentage) that the true population parameter falls within the calculated interval. In this case, our confidence level is 90%.

Calculating a 90% Confidence Interval for the Population Mean (σ Known)

This scenario assumes we know the population standard deviation (σ). This is rarely the case in real-world applications, but it simplifies the calculations and provides a good starting point for understanding the concept.

The formula for a 90% confidence interval for the population mean (μ) when σ is known is:

CI = x̄ ± Z_α/2 * (σ / √n)

Where:

• x̄: The sample mean.
• σ: The population standard deviation.
• n: The sample size.
• Z_α/2: The critical Z-value corresponding to the desired confidence level (90%). α represents the significance level (1 - confidence level), so α = 0.10. Therefore, α/2 = 0.05. The Z-value corresponding to 0.05 in the upper tail of the standard normal distribution is approximately 1.645. You can find this value using a Z-table or statistical software.

Example:

Let's say we have a sample of 100 light bulbs, and we measure their average lifespan (x̄) to be 1000 hours. We know the population standard deviation (σ) is 100 hours. To calculate the 90% confidence interval:

1. Find Z_α/2: As mentioned, for a 90% confidence level, Z_α/2 = 1.645.
2. Calculate the Margin of Error: Margin of Error = 1.645 * (100 / √100) = 16.45 hours.
3. Calculate the Confidence Interval: CI = 1000 ± 16.45 = (983.55 hours, 1016.45 hours)

Therefore, we are 90% confident that the true average lifespan of all light bulbs in the population lies between 983.55 and 1016.45 hours.

Calculating a 90% Confidence Interval for the Population Mean (σ Unknown)

In most real-world situations, the population standard deviation (σ) is unknown. In such cases, we use the sample standard deviation (s) as an estimate. This leads us to use the t-distribution instead of the standard normal distribution.

The formula for a 90% confidence interval for the population mean (μ) when σ is unknown is:

CI = x̄ ± t_{α/2, df} * (s / √n)

Where:

• x̄: The sample mean.
• s: The sample standard deviation.
• n: The sample size.
• t_{α/2, df}: The critical t-value corresponding to the desired confidence level (90%) and degrees of freedom (df = n - 1). You can find this value using a t-table or statistical software.

Example:

Let's assume we have a sample of 25 students, and their average test score (x̄) is 75. The sample standard deviation (s) is 10. To calculate the 90% confidence interval:

1. Determine Degrees of Freedom: df = 25 - 1 = 24
2. Find t_{α/2, df}: For a 90% confidence level and 24 degrees of freedom, the critical t-value (from a t-table or software) is approximately 1.711.
3. Calculate the Margin of Error: Margin of Error = 1.711 * (10 / √25) = 3.422
4. Calculate the Confidence Interval: CI = 75 ± 3.422 = (71.578, 78.422)

We are 90% confident that the true average test score for all students in the population lies between 71.578 and 78.422.

Calculating a 90% Confidence Interval for a Population Proportion

When dealing with proportions (e.g., the percentage of voters who support a particular candidate), the calculation differs slightly. We use the sample proportion (p̂) as an estimate of the population proportion (p). For larger sample sizes, the normal approximation to the binomial distribution is used.

The formula for a 90% confidence interval for a population proportion (p) is:

CI = p̂ ± Z_α/2 * √[(p̂(1 - p̂) / n)]

Where:

• p̂: The sample proportion.
• n: The sample size.
• Z_α/2: The critical Z-value corresponding to the 90% confidence level (1.645).

Example:

Suppose a survey of 500 people reveals that 300 support a particular policy. The sample proportion is p̂ = 300/500 = 0.6. To calculate the 90% confidence interval:

1. Find Z_α/2: Z_α/2 = 1.645
2. Calculate the Margin of Error: Margin of Error = 1.645 * √[(0.6 * 0.4 / 500)] ≈ 0.036
3. Calculate the Confidence Interval: CI = 0.6 ± 0.036 = (0.564, 0.636)

We are 90% confident that the true proportion of the population who support the policy lies between 56.4% and 63.6%.

Factors Affecting Confidence Interval Width

The width of the confidence interval reflects the precision of our estimate. A narrower interval indicates greater precision. Several factors influence the width:

• Sample Size (n): Larger sample sizes lead to narrower intervals. More data reduces uncertainty.
• Population Standard Deviation (σ or s): Larger standard deviations result in wider intervals, indicating greater variability in the data.
• Confidence Level: Higher confidence levels (e.g., 99% instead of 90%) produce wider intervals. Greater confidence requires a larger range.

Choosing the Appropriate Method

Selecting the correct method for calculating a 90% confidence interval depends on the type of data (mean or proportion) and whether the population standard deviation is known. Always carefully consider these factors before proceeding with the calculations.

Interpretation and Limitations

It is crucial to correctly interpret the confidence interval. It's not a statement about the probability of the population parameter being within the specific calculated interval. Instead, it reflects the long-run frequency of intervals containing the true parameter if the sampling process were repeated numerous times.

Conclusion

Calculating a 90% confidence interval is a valuable tool for making inferences about population parameters based on sample data. Understanding the underlying principles, choosing the appropriate method, and correctly interpreting the results are essential for effectively using this statistical technique. Remember to always consider the limitations and potential sources of error when drawing conclusions from confidence intervals. The examples provided offer a practical understanding of the process, allowing you to apply these methods to your own data analysis. Remember to use appropriate statistical software or tools for accurate calculations, especially with larger datasets or more complex scenarios.