How To Compute 90 Confidence Interval

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

Understanding confidence intervals is crucial for anyone working with statistical data. They provide a range of values within which we can be reasonably confident the true population parameter lies. This guide will delve into the specifics of calculating a 90% confidence interval, exploring various scenarios and offering practical examples.

What is a Confidence Interval?

Before diving into the calculations, let's clarify the concept. A confidence interval estimates a range of plausible values for a population parameter (like the mean or proportion) based on a sample of data. The confidence level, 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 means that if we were to repeat the sampling process many times, 90% of the calculated intervals would contain the true population parameter.

Key Components of Confidence Interval Calculation

Calculating a confidence interval involves several key components:

• Sample Data: The data collected from your sample. This could be anything from survey responses to measurements of physical properties.
• Sample Statistic: The relevant statistic calculated from your sample data. This is often the sample mean (x̄) for estimating population mean or sample proportion (p̂) for estimating population proportion.
• Standard Error: A measure of the variability of the sample statistic. It quantifies how much the sample statistic is likely to vary from the true population parameter. The formula for standard error varies depending on the statistic and whether the population standard deviation is known.
• Critical Value: This value is determined by the chosen confidence level and the distribution of the sample statistic (typically a t-distribution or a z-distribution). It represents the number of standard errors away from the sample statistic that defines the interval's boundaries.
• Margin of Error: The product of the critical value and the standard error. This represents the amount added and subtracted from the sample statistic to obtain the confidence interval's upper and lower bounds.

Calculating a 90% Confidence Interval for the Population Mean

The method for calculating a confidence interval depends on whether the population standard deviation (σ) is known or unknown.

1. Population Standard Deviation (σ) is Known: Using the Z-distribution

When the population standard deviation is known, we use the z-distribution to determine the critical value. The formula for the 90% confidence interval is:

Confidence Interval = Sample Mean ± (Z-score * (σ / √n))

Where:

• Sample Mean (x̄): The average of your sample data.
• Z-score: The critical value corresponding to a 90% confidence level. For a two-tailed test, this is approximately 1.645. You can find this value using a z-table or statistical software.
• σ: The population standard deviation.
• n: The sample size.

Example:

Let's say 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:

1. Z-score: The z-score for a 90% confidence interval (two-tailed) is 1.645.
2. Standard Error: Standard Error = σ / √n = 100 / √100 = 10
3. Margin of Error: Margin of Error = Z-score * Standard Error = 1.645 * 10 = 16.45
4. Confidence Interval: Confidence Interval = x̄ ± Margin of Error = 1000 ± 16.45 = (983.55, 1016.45)

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

2. Population Standard Deviation (σ) is Unknown: Using the t-distribution

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

Confidence Interval = Sample Mean ± (t-score * (s / √n))

Where:

• Sample Mean (x̄): The average of your sample data.
• t-score: The critical value from the t-distribution. This value depends on both the confidence level (90%) and the degrees of freedom (n-1). You can find this value using a t-table or statistical software.
• s: The sample standard deviation.
• n: The sample size.

Example:

Suppose we have a sample of 25 students with a mean exam score (x̄) of 75 and a sample standard deviation (s) of 10. To calculate the 90% confidence interval:

1. Degrees of Freedom: Degrees of Freedom = n - 1 = 25 - 1 = 24
2. t-score: Using a t-table or software with 24 degrees of freedom and a 90% confidence level (two-tailed), the t-score is approximately 1.711.
3. Standard Error: Standard Error = s / √n = 10 / √25 = 2
4. Margin of Error: Margin of Error = t-score * Standard Error = 1.711 * 2 = 3.422
5. Confidence Interval: Confidence Interval = x̄ ± Margin of Error = 75 ± 3.422 = (71.578, 78.422)

Therefore, we are 90% confident that the true mean exam score for the entire student 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 people who prefer a certain product), the calculation is slightly different. We use the following formula:

Confidence Interval = p̂ ± (Z-score * √((p̂(1-p̂))/n))

Where:

• p̂: The sample proportion.
• Z-score: The critical value from the z-distribution for a 90% confidence level (approximately 1.645 for a two-tailed test).
• n: The sample size.

Example:

Suppose a survey of 500 people reveals that 300 prefer Brand A. The sample proportion (p̂) is 300/500 = 0.6. To calculate the 90% confidence interval:

1. Z-score: The z-score is 1.645.
2. Standard Error: Standard Error = √((p̂(1-p̂))/n) = √((0.6 * 0.4) / 500) ≈ 0.0219
3. Margin of Error: Margin of Error = Z-score * Standard Error = 1.645 * 0.0219 ≈ 0.036
4. Confidence Interval: Confidence Interval = p̂ ± Margin of Error = 0.6 ± 0.036 = (0.564, 0.636)

We are 90% confident that the true population proportion of people who prefer Brand A lies between 56.4% and 63.6%.

Interpreting Confidence Intervals

It's crucial to understand the correct interpretation of a confidence interval. It does not mean there is a 90% probability that the true population parameter is within the calculated interval. The true parameter is either within the interval or it isn't; the probability is either 100% or 0%. Instead, the 90% refers to the reliability of the method used to construct the interval. If we were to repeat the sampling and calculation process many times, 90% of the resulting intervals would contain the true population parameter.

Factors Affecting Confidence Interval Width

The width of the confidence interval reflects the precision of our estimate. Several factors influence this width:

• Confidence Level: A higher confidence level (e.g., 99% instead of 90%) leads to a wider interval, as we need a larger range to be more certain of capturing the true parameter.
• Sample Size: A larger sample size results in a narrower interval, as larger samples provide more precise estimates.
• Population Variability: Higher variability in the population (larger standard deviation) leads to a wider interval, reflecting the greater uncertainty.

Conclusion

Calculating a 90% confidence interval is a valuable tool for making inferences about a population based on sample data. Understanding the underlying principles, the choice between z-distribution and t-distribution, and the correct interpretation of the results are crucial for effective statistical analysis. Remember to always consider the context of your data and the limitations of your sample when drawing conclusions. By carefully applying these methods, you can gain valuable insights and make informed decisions based on your data.