90 Confidence Interval Z Score

Decoding the 90% Confidence Interval: Understanding Z-Scores and Their Application

The concept of a confidence interval is fundamental in statistics, providing a range of values within which a population parameter is likely to fall with a certain degree of confidence. A common scenario involves calculating a 90% confidence interval, which means we're 90% confident that the true population parameter lies within the calculated range. This article will delve into the specifics of calculating a 90% confidence interval using z-scores, explaining the underlying principles and providing practical examples. We'll also explore the assumptions behind this method and address some frequently asked questions. Understanding this will equip you with a crucial tool for data analysis and interpretation.

Introduction to Confidence Intervals and Z-scores

Before diving into the specifics of a 90% confidence interval, let's establish a foundational understanding. A confidence interval estimates a range of values likely to contain an unknown population parameter (like the population mean or proportion). The level of confidence, expressed as a percentage (e.g., 90%, 95%, 99%), reflects the probability that the interval contains the true parameter.

A z-score, also known as a standard score, indicates how many standard deviations a data point is from the mean of its distribution. It's a crucial element in calculating confidence intervals, particularly when dealing with normally distributed data or large sample sizes where the central limit theorem applies. The z-score allows us to standardize the data and relate it to the standard normal distribution (mean=0, standard deviation=1), facilitating probability calculations.

Calculating a 90% Confidence Interval using Z-scores

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

CI = x̄ ± Zα/2 * (σ/√n)

Where:

• CI represents the confidence interval.
• x̄ is the sample mean.
• Zα/2 is the critical z-score corresponding to the desired confidence level (90% in this case).
• σ is the population standard deviation.
• n is the sample size.

Let's break down each component:

• x̄ (Sample Mean): This is the average of your sample data. It's a straightforward calculation. For example, if your sample is {10, 12, 15, 18, 20}, the sample mean (x̄) is 15.

• σ (Population Standard Deviation): This represents the variability within the population. Ideally, we'd know the population standard deviation, but often, we only have the sample standard deviation (s). When the sample size is large (generally considered n ≥ 30), we can approximate the population standard deviation with the sample standard deviation (s). Using the sample standard deviation introduces a slight margin of error, addressed later.

• n (Sample Size): The number of observations in your sample. A larger sample size generally leads to a narrower confidence interval, providing a more precise estimate.

• Zα/2 (Critical Z-score): This is the key component for establishing the 90% confidence level. Since we want a 90% confidence interval, α (alpha) represents the remaining 10% (1 - 0.90 = 0.10). Because the distribution is two-tailed (meaning there's 5% probability in each tail), α/2 = 0.05. We then consult a z-table or statistical software to find the z-score corresponding to a cumulative probability of 1 - α/2 = 0.95. This z-score is approximately 1.645.

Therefore, for a 90% confidence interval, the formula becomes:

CI = x̄ ± 1.645 * (σ/√n)

Example Calculation

Let's illustrate this with an example. Suppose we're measuring the average height of adult women in a city. We take a sample of 100 women (n=100), and find the sample mean height to be 165 cm (x̄ = 165 cm). Let's assume the population standard deviation is known to be 5 cm (σ = 5 cm). We want to calculate the 90% confidence interval for the average height of all women in the city.

Using the formula:

CI = 165 ± 1.645 * (5/√100) CI = 165 ± 1.645 * 0.5 CI = 165 ± 0.8225 CI = (164.1775 cm, 165.8225 cm)

Therefore, we are 90% confident that the true average height of adult women in this city falls between 164.18 cm and 165.82 cm.

Using Sample Standard Deviation (s)

In most real-world scenarios, the population standard deviation (σ) is unknown. We use the sample standard deviation (s) instead. When the sample size is large (n ≥ 30), the t-distribution converges to the normal distribution. While using the z-score still provides a reasonable approximation, the accuracy slightly decreases. For smaller sample sizes, using the t-distribution is more appropriate.

Understanding the Implications of Sample Size and Confidence Level

The sample size significantly impacts the width of the confidence interval. Larger sample sizes lead to narrower intervals, offering more precise estimations. This is because a larger sample provides more information about the population, reducing uncertainty.

The confidence level also affects the interval's width. Higher confidence levels (e.g., 99%) result in wider intervals. This is because to be more confident that the interval contains the true population parameter, we must expand the range. There's a trade-off between confidence and precision; higher confidence comes at the cost of a wider, less precise interval.

Assumptions and Limitations

The calculation of a confidence interval using z-scores relies on several key assumptions:

• Random Sampling: The sample data must be randomly selected from the population to avoid bias.
• Independence: The observations in the sample should be independent of each other.
• Normality (or Large Sample Size): If the sample size is small (n < 30), the population should be approximately normally distributed. For larger sample sizes, the central limit theorem ensures that the sample mean will be approximately normally distributed, regardless of the population's distribution.

Violation of these assumptions can lead to inaccurate or misleading confidence intervals.

Frequently Asked Questions (FAQ)

Q: What is the difference between a 90%, 95%, and 99% confidence interval?

A: The difference lies in the confidence level. A 90% confidence interval means there's a 90% probability that the true population parameter falls within the calculated range. A 95% interval increases this probability to 95%, and a 99% interval to 99%. The higher the confidence level, the wider the interval will be.

Q: Why do we use z-scores in calculating confidence intervals?

A: Z-scores allow us to standardize the data and use the standard normal distribution's properties for probability calculations. This simplifies the process of determining the critical values needed for the confidence interval calculation.

Q: What if my sample size is small (n < 30)?

A: For small sample sizes, especially when the population standard deviation is unknown, it's more appropriate to use the t-distribution instead of the normal distribution. The t-distribution accounts for the additional uncertainty introduced by estimating the population standard deviation from the sample. You would use a t-score instead of a z-score in the confidence interval formula.

Q: Can I use this method for proportions instead of means?

A: Yes, a similar method can be applied to calculate confidence intervals for population proportions. The formula will differ slightly, using the sample proportion and its standard error.

Q: How do I interpret the confidence interval?

A: A confidence interval provides a range of plausible values for the population parameter. For example, a 90% confidence interval of (10, 20) suggests that we are 90% confident that the true population parameter lies between 10 and 20. It does not mean there's a 90% chance the parameter is within that range. The parameter either is or isn't within the interval; the 90% refers to the reliability of the method used to construct the interval.

Conclusion

Calculating a 90% confidence interval using z-scores is a powerful statistical tool for estimating population parameters. Understanding the formula, its components, and the underlying assumptions is crucial for accurate interpretation and application. Remember that while confidence intervals provide valuable insights, they are only estimates. The precision of the estimate depends on the sample size and the inherent variability within the population. Always consider the limitations and assumptions before drawing conclusions based on your confidence interval calculations. Further exploration into the t-distribution and confidence intervals for proportions will enhance your understanding of statistical inference.