Find The 90 Confidence Interval

Finding the 90% Confidence Interval: A Comprehensive Guide

Determining a confidence interval is a crucial aspect of statistical inference. It allows us to estimate the range within which a population parameter, such as the mean or proportion, is likely to fall with a certain degree of confidence. This article provides a comprehensive guide on how to find the 90% confidence interval, explaining the underlying concepts, different scenarios, and the necessary calculations. Understanding confidence intervals is essential for interpreting data and drawing meaningful conclusions in various fields, from research and healthcare to business and finance.

Introduction to Confidence Intervals

A confidence interval provides a range of values within which a population parameter is likely to lie. Instead of providing a single point estimate, which might be inaccurate due to sampling variability, a confidence interval acknowledges the inherent uncertainty associated with estimating population parameters from sample data. The 90% confidence interval, for example, 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 elements of a confidence interval are:

• Point Estimate: The sample statistic (e.g., sample mean, sample proportion) that serves as the best guess for the population parameter.
• Margin of Error: The amount added and subtracted from the point estimate to create the interval's upper and lower bounds. The margin of error reflects the uncertainty in the estimate.
• Confidence Level: The probability (expressed as a percentage) that the interval contains the true population parameter. This is usually denoted by (1-α), where α is the significance level. For a 90% confidence interval, α = 0.1.
• Critical Value: A value obtained from the appropriate probability distribution (often the z-distribution or t-distribution) that corresponds to the chosen confidence level.

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

When the population standard deviation (σ) is known, we use the z-distribution to calculate the confidence interval. 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 corresponding to the desired confidence level (for a 90% confidence interval, α = 0.1, and α/2 = 0.05. The Z_0.05 value is approximately 1.645).
• σ is the population standard deviation.
• n is the sample size.

Example:

Suppose a researcher wants to find the 90% confidence interval for the average height of adult women in a certain city. A sample of 100 women is taken, and the sample mean height is found to be 5'4" (64 inches) with a known population standard deviation of 3 inches.

1. Find the critical z-value: For a 90% confidence interval, Z_0.05 ≈ 1.645. This value can be found using a z-table or statistical software.

2. Calculate the margin of error: Margin of Error = 1.645 * (3 / √100) = 1.645 * 0.3 = 0.4935 inches.

3. Calculate the confidence interval: CI = 64 ± 0.4935 inches. Therefore, the 90% confidence interval is (63.5065 inches, 64.4935 inches).

This means we are 90% confident that the true average height of adult women in the city lies between 63.51 and 64.50 inches.

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

When the population standard deviation (σ) is unknown, which is more common in practice, we use the t-distribution instead of the z-distribution. The t-distribution accounts for the extra uncertainty introduced by estimating σ from the sample data. The formula becomes:

CI = x̄ ± t_{α/2, n-1} * (s / √n)

Where:

• x̄ is the sample mean.
• t_{α/2, n-1} is the critical t-value with (n-1) degrees of freedom and a significance level of α/2. The degrees of freedom (df) are equal to the sample size minus 1 (n-1).
• s is the sample standard deviation.
• n is the sample size.

Example:

Let's consider a scenario where a teacher wants to determine the 90% confidence interval for the average test scores of their students. A sample of 25 students is selected, and the sample mean score is 75 with a sample standard deviation of 10.

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

2. Find the critical t-value: Using a t-table or statistical software with df = 24 and α/2 = 0.05, we find t_{0.05, 24} ≈ 1.711.

3. Calculate the margin of error: Margin of Error = 1.711 * (10 / √25) = 1.711 * 2 = 3.422.

4. Calculate the confidence interval: CI = 75 ± 3.422. The 90% confidence interval is (71.578, 78.422).

This suggests that the teacher is 90% confident that the true average test score for all students lies between 71.58 and 78.42.

Calculating the 90% Confidence Interval for a Population Proportion

When dealing with proportions (e.g., the percentage of people who prefer a particular product), we use a slightly different formula:

CI = p̂ ± Z_α/2 * √[(p̂(1-p̂))/n]

Where:

• p̂ is the sample proportion.
• Z_α/2 is the critical z-value for the desired confidence level (1.645 for a 90% confidence interval).
• n is the sample size.

Example:

Suppose a marketing team wants to estimate the 90% confidence interval for the proportion of customers who are satisfied with their new product. A survey of 400 customers reveals that 320 are satisfied.

1. Calculate the sample proportion: p̂ = 320/400 = 0.8.

2. Find the critical z-value: Z_0.05 ≈ 1.645.

3. Calculate the margin of error: Margin of Error = 1.645 * √[(0.8 * 0.2) / 400] ≈ 1.645 * 0.02 = 0.0329.

4. Calculate the confidence interval: CI = 0.8 ± 0.0329. The 90% confidence interval is (0.7671, 0.8329).

This indicates that the marketing team is 90% confident that the true proportion of satisfied customers lies between 76.71% and 83.29%.

Factors Affecting Confidence Interval Width

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

• Confidence Level: A higher confidence level (e.g., 99% instead of 90%) results in a wider interval because we need a larger range to be more certain of capturing the true parameter.

• Sample Size: A larger sample size leads to a narrower interval because larger samples provide more precise estimates of the population parameter.

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

Interpreting Confidence Intervals

It's crucial to understand the correct interpretation of a confidence interval. The 90% confidence interval does not mean there is a 90% probability that the true population parameter lies within the calculated interval. Instead, it means that if we were to repeatedly take samples and calculate confidence intervals in the same way, 90% of those intervals would contain the true population parameter. The true parameter is fixed; it's our estimate that's uncertain.

Frequently Asked Questions (FAQs)

Q1: What is the difference between a confidence interval and a prediction interval?

A confidence interval estimates the range of a population parameter, while a prediction interval estimates the range of a future observation from the same population.

Q2: Can I use a confidence interval to prove a hypothesis?

No, a confidence interval provides an estimate of the population parameter, but it doesn't directly test a specific hypothesis. Hypothesis testing uses different statistical methods.

Q3: What if my sample size is very small?

With small sample sizes, the t-distribution is preferred over the z-distribution, even if the population standard deviation is known. For extremely small sample sizes, the confidence interval might be quite wide, reflecting the increased uncertainty.

Q4: How do I choose the appropriate confidence level?

The choice of confidence level depends on the context and the consequences of making an incorrect estimate. Higher confidence levels provide more certainty but come at the cost of wider intervals. 90%, 95%, and 99% are common choices.

Q5: What if my data is not normally distributed?

If your data is significantly non-normal, especially with small sample sizes, the use of the t-distribution might not be appropriate. You might need to consider non-parametric methods or transformations of your data.

Conclusion

Calculating and interpreting confidence intervals is a fundamental skill in statistics. Understanding the different formulas and their applications allows researchers and analysts to draw meaningful conclusions from sample data and estimate population parameters with a specified level of confidence. This guide has provided a thorough explanation of how to find the 90% confidence interval in various scenarios, emphasizing the importance of considering sample size, standard deviation, and the choice of distribution. Remember that the confidence level reflects the long-run performance of the method, not the probability that a specific interval contains the true parameter. Mastering this concept is vital for effective data analysis and informed decision-making across numerous fields.