How To Compute 90 Confidence Interval

How to Compute a 90% Confidence Interval

A confidence interval provides a range of values within which a population parameter is likely to fall, with a certain level of confidence. A 90% confidence interval, for example, means that if you were to repeat the sampling process many times, 90% of the calculated intervals would contain the true population parameter. Understanding how to compute a 90% confidence interval is crucial in statistical analysis, offering a powerful way to express uncertainty and make inferences about a population based on sample data. This comprehensive guide will walk you through the process, covering various scenarios and considerations.

Understanding the Basics: Key Concepts and Terminology

Before diving into the calculations, let's clarify some fundamental concepts:

• Population Parameter: This is the true value of a characteristic you're interested in measuring for the entire population (e.g., the average height of all adult women in a country). We usually don't know the true population parameter and use sample data to estimate it.

• Sample Statistic: This is the equivalent characteristic measured in your sample (e.g., the average height of 100 randomly selected adult women).

• Confidence Level: This represents the probability that the confidence interval contains the true population parameter. A 90% confidence level means there's a 90% probability the interval covers the true value.

• Margin of Error: This is the amount added and subtracted from the sample statistic to create the upper and lower bounds of the confidence interval. A smaller margin of error indicates a more precise estimate.

• Critical Value: This value, derived from the appropriate probability distribution (often the t-distribution or z-distribution), determines the width of the confidence interval. It depends on the confidence level and the sample size.

• Standard Error: This measures the variability of the sample statistic. A smaller standard error suggests a more precise estimate of the population parameter.

Computing a 90% Confidence Interval for the Population Mean

The most common scenario involves calculating a 90% confidence interval for the population mean (μ). The method varies slightly depending on whether the population standard deviation (σ) is known or unknown.

When the Population Standard Deviation (σ) is Known

This scenario is less common in practice, but it's conceptually simpler. We use the z-distribution. The formula is:

CI = x̄ ± z_α/2 * (σ / √n)

Where:

• CI represents the confidence interval.
• x̄ is the sample mean.
• z_α/2 is the critical z-value for a 90% confidence level (α = 0.10). For a two-tailed test, α/2 = 0.05. Using a z-table or statistical software, z_0.05 ≈ 1.645.
• σ is the population standard deviation.
• n is the sample size.

Example: Suppose we have a sample of 100 light bulbs, with a mean lifespan (x̄) of 1000 hours and a known population standard deviation (σ) of 100 hours. To calculate the 90% confidence interval:

CI = 1000 ± 1.645 * (100 / √100) = 1000 ± 16.45

Therefore, the 90% confidence interval is (983.55, 1016.45) hours. We can be 90% confident that the true mean lifespan of all light bulbs in the population falls within this range.

When the Population Standard Deviation (σ) is Unknown

This is the more realistic scenario. Since σ is unknown, we estimate it using the sample standard deviation (s). In this case, we use the t-distribution, which accounts for the added uncertainty from estimating σ. The formula is:

CI = x̄ ± t_{α/2, n-1} * (s / √n)

Where:

• t_{α/2, n-1} is the critical t-value for a 90% confidence level with (n-1) degrees of freedom. You'll need a t-table or statistical software to find this value. The degrees of freedom (df) are n-1.

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

1. Degrees of freedom: df = 25 - 1 = 24
2. Critical t-value: Using a t-table or software for a 90% confidence level (α = 0.10) and 24 degrees of freedom, we find t_{0.05, 24} ≈ 1.711.
3. Confidence interval: CI = 75 ± 1.711 * (10 / √25) = 75 ± 3.422

The 90% confidence interval is (71.578, 78.422). We are 90% confident that the true average test score of all students in the population lies within this range.

Computing a 90% Confidence Interval for Other Parameters

The principles described above can be adapted to calculate confidence intervals for other population parameters:

Confidence Interval for a Population Proportion (p)

This is used when you're interested in estimating the proportion of a population that possesses a certain characteristic. The formula is:

CI = p̂ ± z_α/2 * √[(p̂(1-p̂))/n]

Where:

• p̂ is the sample proportion.
• n is the sample size.

Confidence Interval for the Difference Between Two Population Means

When comparing the means of two populations, you might calculate a confidence interval for the difference between their means. The formula depends on whether the population standard deviations are known or unknown and whether the samples are independent or dependent. More complex formulas and considerations apply in these scenarios.

Confidence Interval for the Difference Between Two Population Proportions

Similar to the difference between means, this involves calculating a confidence interval for the difference between two population proportions. Again, more intricate formulas are involved.

Factors Affecting the Width of the Confidence Interval

Several factors influence the width of the confidence interval, affecting the precision of your estimate:

• Confidence Level: A higher confidence level (e.g., 99% instead of 90%) results in a wider interval, reflecting greater certainty but less precision.

• Sample Size: A larger sample size leads to a narrower interval, improving precision because it reduces the standard error.

• Standard Deviation: A larger standard deviation (either population or sample) results in a wider interval, indicating greater variability in the data.

Choosing the Right Method and Interpreting Results

Selecting the appropriate method for calculating a 90% confidence interval depends heavily on the specific context, including the parameter of interest (mean, proportion, etc.), whether the population standard deviation is known, and the sample size. Always consider the assumptions underlying the chosen method.

Interpreting the results involves understanding that the confidence interval provides a range of plausible values for the population parameter. It does not mean there's a 90% chance the true parameter lies within the calculated interval. Instead, it implies that if you repeatedly sampled and calculated confidence intervals using the same method, 90% of those intervals would contain the true population parameter.

Advanced Considerations and Software Applications

For more complex scenarios, such as those involving small sample sizes, non-normal distributions, or clustered data, more advanced statistical techniques might be necessary. Statistical software packages like R, SPSS, SAS, and Python (with libraries like SciPy and Statsmodels) offer robust functions for calculating confidence intervals and performing more sophisticated statistical analyses. These tools automate the calculations and provide additional diagnostic information.

Mastering the computation of a 90% confidence interval is an essential skill for anyone working with statistical data. It provides a powerful method for making inferences about a population based on sample data, while acknowledging and quantifying the uncertainty involved in the process. Remember to carefully choose the appropriate method based on the specifics of your data and research question and to correctly interpret the resulting confidence interval. By understanding these principles, you can effectively communicate your findings and draw sound conclusions from your analyses.