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. A confidence interval provides a range of values within which a population parameter is likely to fall, with a certain level of confidence. This article will delve into the specifics of constructing a 90% confidence interval, explaining the underlying concepts, different scenarios, and the interpretation of the results.

Understanding Confidence Intervals

Before diving into the construction of a 90% confidence interval, let's solidify the fundamental concepts. A confidence interval is an estimate of an unknown population parameter based on sample data. It's expressed as a range of values, typically written as (lower bound, upper bound). The percentage associated with the interval (e.g., 90%, 95%, 99%) represents the confidence level. This confidence level signifies the long-run proportion of confidence intervals that would contain the true population parameter if we were to repeatedly sample from the population and construct intervals using the same method.

Key Components:

• Confidence Level: The probability that the interval contains the true population parameter. A 90% confidence level means that if we were to repeat the sampling process many times, 90% of the constructed intervals would contain the true population parameter.

• Point Estimate: The sample statistic used to estimate the population parameter. For example, the sample mean (x̄) is used to estimate the population mean (µ).

• Margin of Error: The amount added and subtracted from the point estimate to obtain the confidence interval's upper and lower bounds. It reflects the uncertainty in the estimate due to sampling variability.

• Critical Value: A value from a probability distribution (like the t-distribution or z-distribution) that depends on the confidence level and the degrees of freedom. This value determines the margin of error.

Constructing a 90% Confidence Interval for the Population Mean

The most common type of confidence interval is for the population mean. The method for constructing this interval differs depending on whether the population standard deviation is known or unknown.

When 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 a 90% confidence interval for the population mean (µ) is:

x̄ ± Z_α/2 * (σ / √n)

Where:

• x̄: The sample mean
• Z_α/2: The critical z-value for a 90% confidence level. α = 1 - 0.90 = 0.10, so 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.
• σ: The population standard deviation
• n: The sample size

Example:

Let's say we have a sample of 100 observations with a sample mean (x̄) of 50 and a known population standard deviation (σ) of 10.

1. Find the critical z-value: For a 90% confidence interval, Z_α/2 = 1.645.

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.

When 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:

x̄ ± t_{α/2, df} * (s / √n)

Where:

• x̄: The sample mean
• t_{α/2, df}: The critical t-value for a 90% confidence level with (n-1) degrees of freedom (df). You'll need to consult a t-table or statistical software to find this value.
• s: The sample standard deviation
• n: The sample size

Example:

Suppose we have a sample of 25 observations with a sample mean (x̄) of 75 and a sample standard deviation (s) of 5.

1. Determine the degrees of freedom: df = n - 1 = 25 - 1 = 24

2. Find the critical t-value: For a 90% confidence level and 24 degrees of freedom, the critical t-value (t_{α/2, df}) is approximately 1.711. This value can be found using a t-table or statistical software.

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)

We are 90% confident that the true population mean lies between 73.289 and 76.711.

Constructing a 90% Confidence Interval for Other Parameters

While the population mean is the most frequently estimated parameter, confidence intervals can also be constructed for other population parameters, such as:

Population Proportion (p)

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

p̂ ± Z_α/2 * √[(p̂(1-p̂))/n]

Where:

• p̂: The sample proportion
• Z_α/2: The critical z-value for a 90% confidence level (1.645)
• n: The sample size

Difference Between Two Population Means

When comparing the means of two populations, the construction of the confidence interval depends on whether the population variances are assumed equal or unequal. Different formulas and procedures apply to these scenarios.

Difference Between Two Population Proportions

Similar to comparing means, comparing proportions requires considering different formulas depending on the specific assumptions made.

Factors Affecting Confidence Interval Width

The width of a confidence interval reflects the precision of the estimate. A narrower interval indicates a more precise estimate. Several factors influence the width:

• Confidence Level: Higher confidence levels (e.g., 99% instead of 90%) result in wider intervals because greater confidence requires a larger range to encompass the true parameter.

• Sample Size: Larger sample sizes lead to narrower intervals. Larger samples provide more information about the population, reducing uncertainty.

• Population Variability: Higher population variability (σ or s) leads to wider intervals. More variability means the sample mean is less precise in estimating the population mean.

Interpreting Confidence Intervals

It's crucial to interpret confidence intervals correctly. A 90% confidence interval does not mean there's a 90% chance that the true population parameter falls within the calculated interval. The true parameter is either within the interval or it isn't; the probability is either 1 or 0. Instead, the 90% refers to the long-run frequency of intervals constructed in this way that will contain the true parameter. If we were to repeatedly sample and construct 90% confidence intervals, about 90% of them would contain the true population parameter.

Conclusion

Constructing a 90% confidence interval is a fundamental statistical procedure used to estimate population parameters. Understanding the underlying concepts, choosing the appropriate formula based on the available data and the parameter being estimated, and correctly interpreting the results are crucial for making sound inferences from data. Remember to always consider the limitations of confidence intervals and avoid misinterpretations. Using statistical software can significantly simplify the calculations involved in constructing confidence intervals. However, understanding the underlying principles remains vital for effective data analysis and interpretation.