Z Score Of 1.645

Decoding the Z-Score of 1.645: A Deep Dive into Statistical Significance

The z-score of 1.645 holds a significant place in statistics, particularly in hypothesis testing and determining confidence intervals. Understanding its meaning and application is crucial for interpreting statistical results and drawing meaningful conclusions from data. This article will provide a comprehensive explanation of the z-score of 1.645, its implications, and practical applications, demystifying its importance for students and professionals alike. We will explore its role in various statistical contexts, including calculating confidence intervals and determining statistical significance in hypothesis tests.

Understanding Z-Scores and the Standard Normal Distribution

Before delving into the specifics of a z-score of 1.645, it's essential to grasp the fundamental concept of z-scores and the standard normal distribution. A z-score, also known as a standard score, indicates how many standard deviations a data point is from the mean of a distribution. It's a standardized measure that allows for the comparison of data points from different distributions with varying means and standard deviations.

The standard normal distribution is a theoretical probability distribution with a mean of 0 and a standard deviation of 1. It's a bell-shaped curve, symmetrical around its mean. Transforming raw data into z-scores allows us to visualize the data's position relative to the mean within this standard distribution. This standardization is particularly useful when comparing values from different datasets.

The formula for calculating a z-score is:

z = (x - μ) / σ

where:

• x is the individual data point
• μ is the population mean
• σ is the population standard deviation

A positive z-score indicates that the data point is above the mean, while a negative z-score indicates it's below the mean. A z-score of 0 signifies that the data point is equal to the mean.

The Significance of a Z-Score of 1.645

The z-score of 1.645 holds particular importance because it corresponds to specific probabilities within the standard normal distribution. More specifically, it's closely linked to the one-tailed 5% significance level and the two-tailed 10% significance level.

Let's break this down:

• One-tailed test (5% significance level): In a one-tailed test, we're interested in whether a sample mean is significantly greater than or less than a hypothesized population mean. A z-score of 1.645 corresponds to the upper 5% of the standard normal distribution. This means that there's a 5% probability of observing a z-score of 1.645 or higher if the null hypothesis (that there's no difference between the sample and population means) is true. In simpler terms, if we obtain a z-score greater than or equal to 1.645 in a one-tailed test, we reject the null hypothesis at the 5% significance level. This indicates strong evidence against the null hypothesis.

• Two-tailed test (10% significance level): In a two-tailed test, we're interested in whether a sample mean is significantly different from a hypothesized population mean, regardless of whether it's greater or less. Here, a z-score of 1.645 (or -1.645) corresponds to the 5% probability in one tail. Because we are considering both tails of the distribution (the upper and lower 5%), the combined probability in both tails is 10%. Therefore, if we obtain a z-score greater than or equal to 1.645 or less than or equal to -1.645, we reject the null hypothesis at the 10% significance level.

Applications of the Z-Score of 1.645

The z-score of 1.645 finds practical application in various statistical scenarios:

• Hypothesis Testing: As discussed earlier, it's a critical value used in determining statistical significance in both one-tailed and two-tailed hypothesis tests. If the calculated z-score from a sample exceeds 1.645 (or -1.645 in a two-tailed test), the null hypothesis is rejected, suggesting a statistically significant difference.

• Confidence Intervals: The z-score of 1.645 is used in constructing 90% confidence intervals. A 90% confidence interval provides a range of values within which we are 90% confident that the true population mean lies. The formula for a 90% confidence interval is:

Confidence Interval = Sample Mean ± (1.645 * (Standard Deviation / √n))

where:

• n is the sample size

This means that 90% of the time, the true population mean will fall within this calculated range.

• Determining Probabilities: The z-score of 1.645 can be used to determine probabilities associated with specific regions under the standard normal curve. For example, the probability of observing a z-score greater than 1.645 is 5%, while the probability of observing a z-score between -1.645 and 1.645 is approximately 90%.

• Quality Control: In quality control processes, the z-score can be used to monitor the deviation of a product's characteristics from the desired specifications. A consistently high z-score might indicate a problem in the manufacturing process.

Illustrative Example: Hypothesis Testing

Let's consider a hypothetical scenario to illustrate the application of the z-score of 1.645 in hypothesis testing. Suppose a company claims that its new light bulbs have an average lifespan of 1000 hours. A consumer group tests a sample of 100 light bulbs and finds a mean lifespan of 980 hours with a standard deviation of 80 hours. We want to test the company's claim at a 5% significance level using a one-tailed test.

1. State the Null and Alternative Hypotheses:

   • Null Hypothesis (H0): The average lifespan is 1000 hours (μ = 1000).
   • Alternative Hypothesis (H1): The average lifespan is less than 1000 hours (μ < 1000). This is a one-tailed test.

2. Calculate the Z-score:

   • z = (980 - 1000) / (80 / √100) = -2.5

3. Compare the Z-score to the Critical Value:

   • Our calculated z-score of -2.5 is less than -1.645. (Remember, we are looking at the left tail in a one-tailed test because our alternative hypothesis states the average lifespan is less than 1000).

4. Draw a Conclusion:

   • Since the calculated z-score falls in the rejection region (less than -1.645), we reject the null hypothesis at the 5% significance level. This provides strong evidence that the average lifespan of the light bulbs is less than 1000 hours, contradicting the company’s claim.

Frequently Asked Questions (FAQs)

• What happens if my calculated z-score is exactly 1.645? In most cases, a z-score exactly at the critical value would lead to rejecting the null hypothesis, although some statisticians may prefer to use a more stringent criterion. The most important aspect is understanding the p-value, which provides a more nuanced interpretation.

• Can I use a z-score of 1.645 even if my sample size is small? No. The z-score is based on the assumption of a normal distribution, which is a reasonable assumption for large sample sizes due to the central limit theorem. For small sample sizes, a t-test should be used instead.

• What is the difference between a one-tailed and two-tailed test? A one-tailed test examines whether a sample mean is significantly greater or less than a hypothesized population mean, while a two-tailed test examines whether it's significantly different, regardless of direction.

• How do I find the z-score for other significance levels? You can use a z-table or statistical software to find the z-score corresponding to different significance levels (e.g., 1%, 10%, etc.) for both one-tailed and two-tailed tests.

• Why is the standard normal distribution important? The standard normal distribution provides a common framework for comparing data from different distributions. Its properties make it easy to calculate probabilities and conduct statistical inference.

Conclusion

The z-score of 1.645 serves as a crucial benchmark in statistical analysis, particularly in determining statistical significance and constructing confidence intervals. Understanding its meaning and application within the context of the standard normal distribution is vital for interpreting statistical results accurately. While this article focused on the z-score of 1.645, the principles discussed extend to other z-scores and their roles in statistical inference, emphasizing the importance of a solid grasp of fundamental statistical concepts for effective data analysis and decision-making. Remember to consider the specific context of your analysis, the type of test (one-tailed or two-tailed), and the limitations of using z-scores, particularly for small sample sizes. Utilizing appropriate statistical methods and interpreting results correctly is key to drawing valid conclusions from your data.