How To Find A 90 Confidence Interval

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

Finding a 90% confidence interval is a crucial statistical technique used to estimate a population parameter based on sample data. This guide provides a comprehensive explanation of the process, covering different scenarios and offering insights to help you understand and apply this valuable tool. Understanding confidence intervals is essential for anyone working with data analysis, research, or any field requiring statistical inference. We'll explore the underlying principles, the steps involved, and address common questions, empowering you to confidently calculate and interpret 90% confidence intervals.

Introduction to Confidence Intervals

A confidence interval provides a range of values within which we are confident that the true population parameter lies. A 90% confidence interval, specifically, 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 remember that the confidence interval doesn't mean there's a 90% chance the true parameter is within the calculated interval; the true parameter is either within the interval or it isn't. The 90% refers to the reliability of the method used to construct the interval.

The key elements determining a confidence interval are:

• Sample Data: The data collected from your sample.
• Sample Statistics: Calculated values from your sample data (e.g., sample mean, sample standard deviation).
• Confidence Level: The desired level of confidence (in this case, 90%).
• Critical Value: A value from the appropriate probability distribution (often the t-distribution or z-distribution) corresponding to the chosen confidence level.
• Standard Error: A measure of the variability of the sample statistic.

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

When the population standard deviation (σ) is known, we use the z-distribution to calculate the confidence interval. This scenario is less common in real-world applications, but it provides a foundational understanding.

Steps:

1. Calculate the sample mean (x̄): Sum all the data points in your sample and divide by the sample size (n).

2. Determine the critical value (z):* For a 90% confidence interval, the alpha level (α) is 1 - 0.90 = 0.10. Since we're dealing with a two-tailed test, we divide α by 2, resulting in 0.05. Consult a z-table or use statistical software to find the z-score corresponding to a cumulative probability of 1 - 0.05 = 0.95. This value is approximately 1.645.

3. Calculate the standard error (SE): The standard error is calculated as σ/√n, where σ is the population standard deviation and n is the sample size.

4. Calculate the margin of error (ME): The margin of error is z* * SE.

5. Calculate the confidence interval: The 90% confidence interval is given by: x̄ ± ME. This means the lower bound of the interval is x̄ - ME, and the upper bound is x̄ + ME.

Example:

Suppose we have a sample of n = 25 observations with a sample mean (x̄) of 50 and a known population standard deviation (σ) of 10.

1. x̄ = 50
2. z* = 1.645 (from the z-table for 95th percentile)
3. SE = 10/√25 = 2
4. ME = 1.645 * 2 = 3.29
5. Confidence Interval: 50 ± 3.29 = (46.71, 53.29)

Therefore, we are 90% confident that the true population mean lies between 46.71 and 53.29.

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

In most real-world situations, the population standard deviation is unknown. In such cases, we use the t-distribution instead of the z-distribution. The t-distribution accounts for the additional uncertainty introduced by estimating the population standard deviation from the sample data.

Steps:

1. Calculate the sample mean (x̄) and the sample standard deviation (s): These are calculated from your sample data.

2. Determine the degrees of freedom (df): The degrees of freedom are calculated as n - 1, where n is the sample size.

3. Determine the critical value (t):* Consult a t-table or use statistical software to find the t-score corresponding to a 90% confidence level and the calculated degrees of freedom. Remember to consider the two-tailed nature of the test.

4. Calculate the standard error (SE): The standard error is calculated as s/√n, where s is the sample standard deviation and n is the sample size.

5. Calculate the margin of error (ME): The margin of error is t* * SE.

6. Calculate the confidence interval: The 90% confidence interval is given by: x̄ ± ME.

Example:

Let's say we have a sample of n = 16 observations with a sample mean (x̄) of 75 and a sample standard deviation (s) of 8.

1. x̄ = 75, s = 8
2. df = 16 - 1 = 15
3. t* ≈ 1.753 (from the t-table for 90% confidence and 15 degrees of freedom)
4. SE = 8/√16 = 2
5. ME = 1.753 * 2 = 3.506
6. Confidence Interval: 75 ± 3.506 = (71.494, 78.506)

Therefore, we are 90% confident that the true population mean lies between 71.494 and 78.506.

Calculating a 90% Confidence Interval for a Population Proportion

When dealing with proportions (e.g., the percentage of people who prefer a certain product), we use a different formula based on the binomial distribution. For larger sample sizes, the normal approximation to the binomial distribution can be used.

Steps:

1. Calculate the sample proportion (p̂): This is the number of successes divided by the sample size (n).

2. Calculate the standard error (SE): The standard error for a proportion is √(p̂(1-p̂)/n).

3. Determine the critical value (z):* As with the population mean with known σ, for a 90% confidence interval, z* is approximately 1.645.

4. Calculate the margin of error (ME): The margin of error is z* * SE.

5. Calculate the confidence interval: The 90% confidence interval is given by: p̂ ± ME.

Example:

Suppose in a sample of n = 100 people, 60 prefer a particular brand (p̂ = 0.6).

1. p̂ = 0.6
2. SE = √(0.6 * 0.4 / 100) = 0.049
3. z* = 1.645
4. ME = 1.645 * 0.049 = 0.0805
5. Confidence Interval: 0.6 ± 0.0805 = (0.5195, 0.6805)

We are 90% confident that the true population proportion lies between 0.5195 and 0.6805.

Interpreting Confidence Intervals

The interpretation of a confidence interval is crucial. A 90% confidence interval of (a, b) indicates that we are 90% confident that the true population parameter lies within the interval (a, b). It does not mean there is a 90% probability that the parameter is within this specific interval. The probability refers to the long-run performance of the method used to construct the interval. If you were to repeat the sampling and interval calculation many times, approximately 90% of the resulting intervals would contain the true population parameter.

Factors Affecting Confidence Interval Width

The width of a confidence interval is influenced by several factors:

• Sample Size (n): Larger sample sizes lead to narrower intervals, providing more precise estimates.
• Confidence Level: Higher confidence levels (e.g., 99% instead of 90%) result in wider intervals. Greater confidence requires a wider range to encompass the true parameter with higher certainty.
• Standard Deviation (σ or s): Larger standard deviations lead to wider intervals, reflecting greater variability in the data.

Frequently Asked Questions (FAQ)

Q: What is the difference between a confidence interval and a prediction interval?

A: A confidence interval estimates the population parameter, while a prediction interval estimates the range where a future observation from the population is likely to fall. Prediction intervals are generally wider than confidence intervals.

Q: Can I use a 90% confidence interval for all types of data?

A: No, the appropriate method for calculating a confidence interval depends on the type of data (e.g., continuous, categorical) and the parameter being estimated (e.g., mean, proportion).

Q: How do I choose the appropriate confidence level?

A: The choice of confidence level depends on the context and the consequences of making an incorrect inference. Higher confidence levels offer greater assurance but at the cost of wider intervals. A 90% confidence level is a common choice, balancing precision and confidence.

Q: What if my sample size is very small?

A: With small sample sizes, the t-distribution should be used even if estimating the population mean. The accuracy of the interval estimate is affected by small sample sizes and might lead to wider intervals.

Conclusion

Calculating and interpreting confidence intervals is a fundamental skill in statistics. A 90% confidence interval provides a valuable range of plausible values for a population parameter based on sample data. Understanding the underlying principles, the steps involved for different scenarios, and the factors that influence the interval's width empowers you to use this tool effectively in various applications. Remember to always choose the appropriate method based on your data and the parameter of interest and carefully consider the implications of the confidence level you select. By mastering this technique, you can significantly enhance your ability to draw meaningful conclusions from data and make informed decisions based on statistical evidence.