Problem 62 - Thinking Mathematically

Question

A set of data items is normally distributed with a mean of 400 and a standard deviation of 50. In Exercises \(59-66\), find the data item in this distribution that corresponds to the given z-score. \(z=2.5\)

Step-by-Step Solution

Answer

The data item in this distribution that corresponds to the given z-score 2.5 is 525.

Step 1: Identify the given values

From the problem, we know that the mean (μ) is 400, the standard deviation (σ) is 50, and the z-score (Z) is 2.5.

Step 2: Substitute the values into the z-score formula

Apply these values into the z-score formula \(X = μ + Zσ\) which becomes \(X = 400 + 2.5(50)\).

Step 3: Compute for X

Multiplying 2.5 and 50 gives 125. Then add 125 to 400, which gives 525. Therefore, the score corresponding to the z-score 2.5 is 525.

Key Concepts

Understanding the Z-score

The z-score is a statistical measure that describes a data point's relationship to the mean of a group of data points. It essentially tells us how many standard deviations away a particular value is from the mean. This is particularly useful when dealing with normally distributed data.

• A positive z-score indicates the data point is above the mean.
• A negative z-score shows it is below the mean.
• A z-score of zero means the data point is exactly at the mean.

To calculate the z-score, the formula is given by \( Z = \frac{X - \mu}{\sigma} \), where

• \(X\) is the data point,
• \(\mu\) is the mean of the data set, and
• \(\sigma\) is the standard deviation.

For example, in the exercise you provided, we were given a z-score of 2.5. This means the particular data item in question is 2.5 standard deviations above the mean.

The Mean in Normal Distribution

The mean is one of the most essential concepts in statistics. It represents the average of a set of numbers and serves as a central value. In a normal distribution, the mean is also the peak of the bell curve. This is because data is symmetrical around the mean.

• To find the mean of a data set, simply add together all data points and divide by the number of points.
• The mean gives you a quick snapshot of the data's center.

In our exercise, the mean is 400. This means that, on average, the data items cluster around the value of 400. When we calculate for a z-score, we use this mean value as a reference point to understand how far a specific data item is from this center.

Role of Standard Deviation

Standard deviation is a measure that indicates the amount of variability or dispersion in a set of data. It tells us how spread out the numbers are around the mean. In a normal distribution, data is spread symmetrically around the mean, and standard deviation helps quantify that spread.

• If standard deviation is small, data points are closely clustered around the mean.
• If it's large, data is more spread out.

The formula for standard deviation is \( \sigma = \sqrt{\frac{\sum (X - \mu)^2}{N}} \), where

• \(X\) represents each data point,
• \(\mu\) is the mean, and
• \(N\) is the number of data points.

In our context, a standard deviation of 50 means most data points in our set will fall within 50 units of 400. With our z-score of 2.5, we use this standard deviation to find how far, or how different, a specific data point is from the average.

Applying Data Analysis

Data analysis involves examining datasets to extract meaningful insights or predictions. It often uses concepts like z-score, mean, and standard deviation to transform raw data into useful information. Especially with normally distributed data, these statistics help us understand patterns and anomalies within the data set.

• We use the z-score and standard deviation to pinpoint the relative standing of a data item.
• The mean provides a baseline for evaluating other statistics.
• By applying these tools, analysts can describe and predict behavior under normal distribution conditions efficiently.

In the task at hand, using data analysis tools allowed us to find that a data point with a z-score of 2.5 in our set, reflecting a score value of 525, signifies it's substantially higher than the average score of 400. Understanding these connections is key in effective data analysis.