90 Confidence Interval Z Score

Decoding the 90% Confidence Interval: A Deep Dive into Z-Scores and Their Applications

Understanding confidence intervals is crucial for anyone working with statistical data, whether you're a seasoned researcher, a data analyst, or simply someone curious about interpreting statistical findings. This article delves into the specifics of a 90% confidence interval, focusing on the role of z-scores in its calculation and practical interpretation. We'll explore the underlying concepts, the step-by-step calculation process, common applications, and the limitations of this statistical tool. By the end, you'll have a firm grasp of what a 90% confidence interval represents and how to effectively use it in your analyses.

What is a Confidence Interval?

A confidence interval provides a range of values within which a population parameter is likely to fall, with a certain degree of confidence. Imagine you're trying to estimate the average height of all adult women in a country. Instead of measuring every single woman, you take a sample and calculate the average height of that sample. This sample average will likely be slightly different from the true population average. A confidence interval accounts for this sampling error by providing an interval around the sample average, indicating the plausible range for the true population average.

The confidence level expresses the probability that the interval contains the true population parameter. A 90% confidence interval means that if you were to repeat the sampling process many times, 90% of the calculated intervals would contain the true population parameter. It's crucial to understand that this doesn't mean there's a 90% chance the true population parameter lies within this specific 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 Role of Z-Scores in 90% Confidence Intervals

Z-scores, also known as standard scores, measure how many standard deviations a data point is from the mean of its distribution. They are essential for constructing confidence intervals when the population standard deviation is known or the sample size is large enough (generally considered to be n ≥ 30) to approximate the population standard deviation with the sample standard deviation. This approximation relies on the Central Limit Theorem.

In a 90% confidence interval, we're interested in the z-score that corresponds to the middle 90% of the standard normal distribution. Because the normal distribution is symmetrical, we need to find the z-score that leaves 5% in each tail (10% total outside the interval). Consulting a z-table or using statistical software, we find that the z-score corresponding to the 95th percentile (leaving 5% in the upper tail) is approximately 1.645. Therefore, the z-score used for a 90% confidence interval is ±1.645. The negative value accounts for the lower 5% tail.

Calculating a 90% Confidence Interval

The formula for calculating a 90% confidence interval for a population mean (µ) when the population standard deviation (σ) is known is:

CI = x̄ ± Z * (σ / √n)

Where:

• CI represents the confidence interval.
• x̄ is the sample mean.
• Z is the z-score corresponding to the desired confidence level (1.645 for 90%).
• σ is the population standard deviation.
• n is the sample size.

Step-by-Step Example:

Let's say we're interested in the average weight of a certain breed of dog. We take a sample of 50 dogs (n=50) and find their average weight to be 25 kg (x̄=25 kg). We know from previous research that the population standard deviation is 3 kg (σ=3 kg). To calculate the 90% confidence interval:

1. Identify the z-score: For a 90% confidence interval, Z = 1.645.

2. Calculate the standard error: Standard error (SE) = σ / √n = 3 / √50 ≈ 0.424

3. Calculate the margin of error: Margin of error (ME) = Z * SE = 1.645 * 0.424 ≈ 0.7

4. Calculate the confidence interval: CI = x̄ ± ME = 25 ± 0.7 = (24.3 kg, 25.7 kg)

Therefore, we can be 90% confident that the true average weight of this breed of dog lies between 24.3 kg and 25.7 kg.

When the Population Standard Deviation is Unknown:

When the population standard deviation is unknown, we use the sample standard deviation (s) as an estimate. However, this requires using the t-distribution instead of the z-distribution. The t-distribution accounts for the additional uncertainty introduced by estimating the standard deviation from the sample. The degrees of freedom for the t-distribution is (n-1), where n is the sample size. The critical t-value for a 90% confidence interval will depend on the sample size (degrees of freedom).

Interpreting the 90% Confidence Interval

The interpretation of a 90% confidence interval is crucial. It does not mean there's 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 using the same method, 90% of those intervals would contain the true population parameter. The interval provides a range of plausible values for the parameter, reflecting the uncertainty associated with estimating it from a sample.

Applications of 90% Confidence Intervals

90% confidence intervals are widely used across various fields:

• Medical Research: Determining the effectiveness of a new drug or treatment by comparing the outcomes in a treatment group to a control group.
• Market Research: Estimating the proportion of consumers who prefer a particular product or brand.
• Environmental Science: Estimating the average concentration of a pollutant in a water body.
• Engineering: Assessing the reliability of a product or system by estimating the mean time to failure.
• Social Sciences: Estimating the prevalence of a particular social behavior or attitude within a population.

Limitations of 90% Confidence Intervals

While valuable, 90% confidence intervals have limitations:

• Assumption of Normality: The calculations assume that the data is approximately normally distributed. If this assumption is violated, the results might be inaccurate. Transformations or non-parametric methods might be necessary.
• Sample Size: The accuracy of the confidence interval depends on the sample size. Larger samples generally lead to narrower intervals, providing a more precise estimate.
• Confidence Level: The choice of confidence level (90% in this case) is arbitrary. Higher confidence levels (e.g., 95% or 99%) lead to wider intervals, reflecting greater uncertainty. The selection of the confidence level depends on the context and the desired level of certainty.
• Margin of Error: The margin of error is directly affected by the sample size and standard deviation. Larger standard deviations and smaller samples lead to larger margins of error, widening the confidence interval.

Conclusion:

The 90% confidence interval, utilizing the z-score of ±1.645 when the population standard deviation is known, is a powerful tool for estimating population parameters. Understanding its calculation, interpretation, and limitations is essential for drawing valid conclusions from statistical data. Remember to consider the underlying assumptions and context when applying and interpreting confidence intervals. Always assess the sample size, the standard deviation, and the nature of your data to ensure you are using the appropriate statistical method and correctly interpreting the resulting confidence interval. By mastering this fundamental concept, you'll significantly enhance your ability to analyze data and make informed decisions based on statistical evidence.