From standard normal tables, P(Z > 1.67) ≈ 0.0475.
["Understanding the Standard Normal Table: How to Find P(Z > 1.67) ≈ 0.0475", "When working with normal distributions in statistics, one of the most essential tools is the standard normal table (also known as the Z-table). Understanding how to use this table enables researchers, students, and data analysts to quickly determine probabilities associated with Z-scores—deviations from the mean in a standardized normal distribution.", "### What is the Standard Normal Distribution?", "The standard normal distribution has a mean (μ) of 0 and a standard deviation (σ) of 1. Because of its symmetry and standardized scale, it serves as a reference for converting any normal distribution to a common scale using the Z-score formula:\n[\nZ = \frac{X - \mu}{\sigma}\n]\nThis transformation allows us to find probabilities like P(Z > 1.67), which are critical in hypothesis testing, confidence intervals, and confidence analysis.", "---", "### How to Use the Standard Normal Table to Find P(Z > 1.67)", "To determine P(Z > 1.67), we use the standard normal table in two key steps:", "1. Find the cumulative probability P(Z ≤ 1.67):\n Look up the Z-score of 1.67 in the Z-table. The value you find corresponds to the area under the standard normal curve from negative infinity to Z = 1.67.\n [\n P(Z \leq 1.67) \approx 0.9525\n ]\n This means 95.25% of the values fall below Z = 1.67.", "2. Calculate the tail probability:\n Since the total area under the normal curve is 1, the probability of Z being greater than 1.67 is:\n [\n P(Z > 1.67) = 1 - P(Z \leq 1.67) = 1 - 0.9525 = 0.0475\n ]\n Thus, P(Z > 1.67) ≈ 0.0475, or about 4.75%.", "---", "### Practical Applications of P(Z > 1.67 ≈ 0.0475", "This probability is valuable in statistical inference. For example:", "- In hypothesis testing, a Z-score of 1.67 corresponds to a one-tailed p-value of 0.0475. If your significance level (α) is 0.05, rejecting the null hypothesis is justified.\n- In quality control or finance, knowing such tail probabilities helps assess rare event risks.\n- It supports building confidence intervals and assessing data outliers relative to the standard normal distribution.", "---", "### Why Use the Z-Table Specifically?", "The standard normal table provides precise cumulative probabilities for Z-scores, enabling accurate computation of tail areas like P(Z > 1.67). While technology can approximate these values, manually referencing Z-tables strengthens foundational statistical understanding and validates computational results.", "---", "### Summary", "- The Z-score of 1.67 corresponds to a cumulative probability of approximately 0.9525.\n- Therefore, P(Z > 1.67) ≈ 1 − 0.9525 = 0.0475.\n- This step-by-step lookup using standard normal tables is a vital skill for probability analysis, statistical hypothesis testing, and data-driven decision making.", "Mastering this calculation empowers more confident interpretation of statistical data and instruments research across scientific and real-world applications.", "---", "Keywords:** standard normal table, Z-score, P(Z > 1.67), normal distribution probability, Z-table lookup, statistical analysis, tail probability, hypothesis testing, normal distribution Z-table."]
