90 Confidence Interval Z Score

Understanding the 90% Confidence Interval and its Z-Score: A Comprehensive Guide

The 90% confidence interval is a crucial concept in statistics, used to estimate a population parameter based on sample data. It essentially provides a range of values within which we are 90% confident the true population parameter lies. Understanding this concept, particularly the role of the z-score, is vital for interpreting statistical results and making informed decisions across various fields, from medical research to market analysis. This article will provide a comprehensive explanation of the 90% confidence interval, its associated z-score, and how to calculate and interpret it.

What is a Confidence Interval?

Before diving into the specifics of a 90% confidence interval, let's clarify the general concept. A confidence interval is a range of values that is likely to contain the true value of an unknown population parameter. This parameter could be anything from the population mean (average) to the population proportion (percentage). The interval is constructed based on a sample of data drawn from the population. The key thing to remember is that we are not saying there's a 90% chance the true value is within the interval; rather, we are 90% confident that the method used to construct the interval will produce an interval containing the true value in repeated sampling.

This seemingly subtle difference is important. The confidence level (in this case, 90%) reflects the reliability of the method, not the probability that the specific interval calculated contains the true value.

The 90% Confidence Interval: The Basics

A 90% confidence interval implies that if we were to repeat the sampling process many times and construct a confidence interval for each sample, approximately 90% of those intervals would contain the true population parameter. The remaining 10% would not. This is a long-run property of the method.

The width of the confidence interval is directly related to the sample size and the variability of the data. Larger sample sizes generally lead to narrower intervals, indicating a more precise estimate. Similarly, data with less variability results in narrower intervals.

Calculating the 90% Confidence Interval for the Population Mean

The formula for calculating a 90% confidence interval for the population mean (µ) is:

Confidence Interval = Sample Mean (x̄) ± (Z-score * Standard Error)

Let's break down each component:

• Sample Mean (x̄): This is the average of the values in your sample.
• Z-score: This is the critical value corresponding to the desired confidence level. For a 90% confidence interval, the z-score is approximately 1.645. We'll delve deeper into how this value is obtained in the next section.
• Standard Error: This measures the variability of the sample mean. It's calculated as the sample standard deviation (s) divided by the square root of the sample size (n): Standard Error = s / √n

Determining the Z-score for a 90% Confidence Interval

The z-score is a crucial element in calculating the confidence interval. It represents the number of standard deviations a given value is from the mean in a standard normal distribution (a bell-shaped curve with a mean of 0 and a standard deviation of 1).

To find the z-score for a 90% confidence interval, we need to consider the area under the standard normal curve. Since the confidence interval is two-tailed (meaning it extends both above and below the sample mean), we need to account for the 10% that lies outside the interval. This means 5% is in each tail (10%/2 = 5%).

We then find the z-score corresponding to the cumulative probability of 0.95 (1 - 0.05 = 0.95). This can be done using a z-table (a statistical table providing probabilities for different z-scores), a statistical calculator, or statistical software. Looking up 0.95 in a z-table will give you a z-score of approximately 1.645. This signifies that 90% of the data in a standard normal distribution lies within ±1.645 standard deviations of the mean.

Example Calculation: 90% Confidence Interval

Let's say we have a sample of 100 students' test scores. The sample mean (x̄) is 75, and the sample standard deviation (s) is 10. To calculate the 90% confidence interval:

1. Calculate the Standard Error: Standard Error = 10 / √100 = 1
2. Calculate the Margin of Error: Margin of Error = 1.645 * 1 = 1.645
3. Calculate the Confidence Interval: Confidence Interval = 75 ± 1.645 = (73.355, 76.645)

Therefore, we are 90% confident that the true average test score for the entire student population lies between 73.355 and 76.645.

Understanding the Implications of the Confidence Level

The choice of confidence level (90%, 95%, 99%, etc.) reflects the desired level of certainty. A higher confidence level (e.g., 99%) results in a wider interval, indicating greater certainty but less precision. A lower confidence level (e.g., 90%) yields a narrower interval, offering greater precision but less certainty. The optimal confidence level depends on the context and the trade-off between precision and certainty. In many scenarios, a 90% confidence interval strikes a reasonable balance.

Factors Affecting the Width of the Confidence Interval

Several factors influence the width of the confidence interval:

• Sample Size (n): A larger sample size generally leads to a narrower confidence interval, providing a more precise estimate. This is because larger samples provide more information about the population.
• Sample Standard Deviation (s): A smaller sample standard deviation leads to a narrower confidence interval. This is because less variability in the data suggests a more precise estimate of the population mean.
• Confidence Level: As mentioned earlier, a higher confidence level leads to a wider interval, while a lower confidence level results in a narrower interval.

The 90% Confidence Interval for Population Proportions

The calculation for the 90% confidence interval for population proportions (p) is slightly different:

Confidence Interval = Sample Proportion (p̂) ± (Z-score * Standard Error)

Where:

• Sample Proportion (p̂): The proportion of successes in the sample.
• Standard Error: √[(p̂(1-p̂))/n]

The z-score for a 90% confidence interval remains 1.645.

Frequently Asked Questions (FAQs)

Q: What does it mean if my 90% confidence interval does not contain the true population parameter?

A: It simply means that in this particular sample, the method did not produce an interval that contained the true population parameter. This is expected to happen in about 10% of cases when using a 90% confidence interval. It doesn't invalidate the method; it highlights the inherent uncertainty in statistical inference.

Q: Can I use a different z-score for a 90% confidence interval?

A: While slight variations may occur depending on the z-table or software used, the z-score of 1.645 is the standard approximation for a 90% two-tailed confidence interval. Using a significantly different value would compromise the accuracy of your interval.

Q: What if my sample size is very small?

A: With small sample sizes, the t-distribution should be used instead of the z-distribution to calculate the confidence interval. The z-distribution assumes a large sample size. The t-distribution accounts for the additional uncertainty associated with small samples.

Q: How do I choose the appropriate confidence level?

A: The choice of confidence level depends on the context and the consequences of being wrong. Higher confidence levels provide greater certainty but reduce precision. Lower confidence levels increase precision but reduce certainty. Consider the potential risks and benefits involved in making a decision based on your confidence interval.

Conclusion

The 90% confidence interval, with its associated z-score of 1.645, is a powerful statistical tool for estimating population parameters. Understanding how to calculate and interpret these intervals is essential for anyone working with data. Remember, the confidence interval doesn't provide a definitive answer, but rather a range of plausible values for the population parameter, allowing us to make informed decisions based on the available data while acknowledging the inherent uncertainty in statistical inference. By understanding the underlying principles and nuances, you can effectively utilize confidence intervals to draw meaningful conclusions from your data. This includes recognizing the impact of sample size, variability, and the confidence level itself on the precision and reliability of your estimates.