Problem 46 - Thinking Mathematically

Question

A set of data items is normally distributed with a mean of 60 and a standard deviation of 8 . In Exercises 33-48, convert each data item to a z-score. 40

Step-by-Step Solution

Answer

The z-score of the data item 40 is -2.5.

Step 1: Identify the Variables

From the problem, we know that the mean (µ) is 60, the standard deviation (σ) is 8, and the raw score (X) that we want to convert is 40.

Step 2: Apply the Z-Score Formula

Put these values into the z-score formula: Z=(X-µ)/σ. So, Z = (40-60)/8.

Step 3: Compute the Z-Score

After doing the math, we find that Z = -2.5

Key Concepts

Understanding the Normal Distribution

The normal distribution is a fundamental concept in statistics often referred to as the bell curve. Its shape is symmetric and depicts how data points tend to cluster around a mean value, with values less likely to occur as they move further away from the mean. In a perfectly normal distribution, the mean, median, and mode of the dataset are equal, with the data evenly distributed around the mean.

In real-world situations, it is used to model a variety of phenomena such as test scores, height, blood pressure, and much more. The normal distribution is crucial because it allows statisticians to make inferences about populations using sample data.

Decoding Standard Deviation

Standard deviation is a measure that quantifies the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation means the values are spread out over a wider range. It is represented symbolically as \(\sigma \) when referring to a population and \(s \) for a sample.

To calculate the standard deviation, one typically follows these steps: find the mean of the data set, subtract the mean from each data point and square the result, compute the average of these squared differences, and finally take the square root of this average. This value helps in understanding the spread of data in relation to the mean.

Mean: The Center of Data

The mean, often called the average, is a measure of central tendency in a data set and is calculated by adding up all the values and then dividing by the number of values. It's symbolized by \(\mu \) when referring to a population and \(\bar{x} \) for a sample mean. The mean is heavily used because it takes into account every value in the dataset, making it a good measure of the center.

However, the mean can be affected by outliers—values that are significantly higher or lower than the majority of the values in the data set, which can skew the results. Therefore, alongside the mean, other measures such as median and mode are also examined for a more comprehensive statistical analysis.

Statistical Analysis: Making Sense of Data

Statistical analysis encompasses a range of techniques for exploring, describing, and inferring conclusions from data. It helps in understanding trends, patterns, and relationships within the data. Through statistical analysis, one can test hypotheses, validate experiments, and make predictions.

Central to this process is the concept of z-score, which standardizes different data points, allowing for comparison across different datasets. A z-score calculates how many standard deviations a data point is from the mean. It is defined as \( Z = (X-\mu)/\sigma \) where \( X \) is a single data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. In the given exercise, the z-score calculation determined that a score of 40 is 2.5 standard deviations below the mean, providing a way to understand its position relative to the rest of the data set.