Finding a z-Score In Exercises 1–16, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.


P33

Verified step by step guidance

1. Identify the given information: The problem states that we need to find the z-score corresponding to the 33rd percentile (P33). This means the cumulative area under the standard normal curve to the left of the z-score is 0.33.
2. Understand the relationship between cumulative area and z-scores: The cumulative area represents the probability that a randomly selected value from the standard normal distribution is less than or equal to the z-score. We will use the Standard Normal Table or technology to find the z-score corresponding to this cumulative area.
3. Use the Standard Normal Table: Locate the cumulative area of 0.33 in the table. The table provides cumulative probabilities for z-scores. Find the row and column that correspond to the closest value to 0.33. The intersection of the row and column gives the z-score.
4. Alternatively, use technology: If using a statistical calculator or software, input the cumulative area (0.33) into the inverse normal function (often labeled as 'invNorm' or similar). Ensure the mean is set to 0 and the standard deviation is set to 1, as this is a standard normal distribution.
5. Interpret the result: The z-score you find represents the point on the standard normal distribution where 33% of the data lies to the left. This z-score can be used for further analysis or interpretation in the context of the problem.

Key Concepts

Z-Score

A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It indicates how many standard deviations an element is from the mean. A positive z-score indicates the value is above the mean, while a negative z-score indicates it is below. Z-scores are essential for standardizing scores on different scales, allowing for comparison across different datasets.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. It is represented by the bell-shaped curve and is used to determine probabilities and percentiles for normally distributed data. The area under the curve corresponds to probabilities, making it a fundamental concept in statistics for understanding how data is distributed.

Cumulative Area

Cumulative area refers to the total area under the curve of a probability distribution up to a certain point. In the context of the standard normal distribution, it represents the probability that a randomly selected score will fall below a specific z-score. This concept is crucial for interpreting z-scores in terms of percentiles, allowing statisticians to understand the relative standing of a score within a distribution.