Problem 55 - Calculus and its applications

Question

Find the \(z\) -value that corresponds to each percentile for a standard normal distribution. a) 30 th percentile b) 50 th percentile c) 95 th percentile

Step-by-Step Solution

Answer

a) -0.52, b) 0, c) 1.645

Step 1: Understanding Percentiles and the Standard Normal Distribution

A percentile is a measure used in statistics that indicates the value below which a given percentage of observations in a group of observations falls. For example, the 30th percentile is the value below which 30% of the observations may be found. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

Step 2: Using the Standard Normal (Z) Table or Calculator

To find the z-values corresponding to the given percentiles, we use the standard normal distribution table (Z-table) or a statistical calculator. These tables or calculators give the probability that a standard normal random variable is less than a given z-value.

Step 3: Finding the z-value for the 30th Percentile

Locate 0.30 (the probability corresponding to the 30th percentile) in the Z-table. The closest value to 0.30 is usually around -0.52, meaning that the 30th percentile corresponds to a z-value of about -0.52.

Step 4: Finding the z-value for the 50th Percentile

The 50th percentile corresponds to the median of the standard normal distribution, which is 0. Hence, the z-value for the 50th percentile is 0.

Step 5: Finding the z-value for the 95th Percentile

Locate 0.95 (the probability corresponding to the 95th percentile) in the Z-table. The value closest to 0.95 typically corresponds to a z-value of about 1.645. This means the z-value associated with the 95th percentile is approximately 1.645.

Key Concepts

Understanding Percentiles

Percentiles are a fundamental concept in statistics, helping to interpret the distribution of data within a set. They represent the position of a data point in relation to the entire data set.
For instance, if you receive a score that falls in the 30th percentile, it means you performed better than or equal to 30% of the participants. Meanwhile, 70% scored better than you.
Percentiles are crucial for understanding data distributions, allowing us to determine how single values compare to the overall dataset.

• 0th Percentile: The minimum value.
• 50th Percentile: The median value.
• 100th Percentile: The maximum value.

Typically, data that follows a standard normal distribution is used to determine percentiles, as it provides a framework where the mean is zero, and the standard deviation is one.

Z-Value Calculation

The z-value, or z-score, represents a standard score in statistics indicating the number of standard deviations a data point is from the mean. Calculating the z-value helps understand where a particular value sits within a normal distribution.
Calculating the z-value involves:

• Identifying the data value of interest.
• Subtracting the mean of the data from this value.
• Dividing by the standard deviation.

Using the formula: \\( z = \frac{(X - \mu)}{\sigma} \)
Where:

• \(X\) is the data value.
• \(\mu\) is the mean of the distribution.
• \(\sigma\) is the standard deviation.

For standard normal distributions, values align directly with percentiles due to the specific mean and standard deviation values.

Using the Normal Distribution Table

A normal distribution table, commonly known as the Z-table, is an invaluable tool for statistics. It provides the probability that a statistic is less than or equal to a given z-value in a standard normal distribution.
Using the Z-table helps in navigating through different percentiles. The table is often divided into two parts:

• The left column usually represents z-values.
• The top row often provides hundredths of z-scores.

To find a specific z-value corresponding to a percentile, trace the percentile value in the table. The intersection of the relevant row and column will guide you to the z-value.
For instance, a 0.95 probability (or 95th percentile) might correlate with a z-value of approximately 1.645. This z-value reveals that 95% of the data falls below this point, ideal for scenarios ranging from assessing scores to setting statistical thresholds.