Problem 18 - Algebra 2 and Trigonometry

Question

The heights of a group of women are normally distributed with a mean of 170 centimeters and a standard deviation of 10 centimeters. What is the \(z\) -score of a member of the group who is 165 centimeters tall?

Step-by-Step Solution

Answer

The z-score is -0.5.

Step 1: Understand the Z-score Formula

The Z-score formula is used to determine how many standard deviations a data point is from the mean. It is given by the formula: \( z = \frac{X - \mu}{\sigma} \), where \( X \) is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

Step 2: Identify Given Values

From the problem statement, identify the given values: the mean \( \mu = 170 \) cm, the standard deviation \( \sigma = 10 \) cm, and the individual height \( X = 165 \) cm.

Step 3: Substitute Values into the Z-score Formula

Substitute the identified values into the Z-score formula: \( z = \frac{165 - 170}{10} \).

Step 4: Calculate the Numerator

Calculate the difference in the numerator: \( 165 - 170 = -5 \).

Step 5: Divide by the Standard Deviation

Divide the result from Step 4 by the standard deviation: \( z = \frac{-5}{10} = -0.5 \).

Key Concepts

Understanding the Normal Distribution

The normal distribution is a fundamental concept in statistics often represented by a bell-shaped curve. This curve is symmetrical, meaning it's perfectly balanced on either side. Most data points cluster around a central peak, the mean, while fewer are found further away on the tails.

• The curve is symmetrical, meaning each half is a mirror image of the other.
• The mean, median, and mode of this distribution are all the same.
• This type of distribution is important as many real-world phenomena tend to approximate a normal distribution.

In our exercise, the heights of women follow this pattern. Understanding that this data is normally distributed gives us the ability to use certain statistical methods, like Z-scores, to understand relative data point positions.

Mean and Standard Deviation Simplified

In statistics, the mean is the average value and is a key indicator of a central point in a data set. When calculating it, you sum up all values and divide by the number of values. The standard deviation, on the other hand, tells us how much the data points deviate from the mean.

• The mean offers a simple, easy-to-understand summary of what a typical data point might be.
• Standard deviation indicates the spread or variability of your data.
• For example, a small standard deviation means that the data points are close to the mean.

In the initial problem, the mean height was 170 centimeters, with a standard deviation of 10 centimeters. These figures set the groundwork for determining how typical or atypical a data point, like the height of 165 cm, is.

Breaking Down Data Point Analysis

Analyzing data points is crucial in statistics to understand individual values in relation to the entire dataset. This requires contextualizing a single data point, like a height of 165 cm, against the overall distribution.

• Data point analysis helps identify if an observation is typical or unusual compared to the group.
• A data point's relative position to the mean provides insights into its uniqueness.

In the Z-score calculation exercise, this concept was used to find out how the height of 165 cm relates to the mean of 170 cm. This analysis allows us to compute the Z-score, which shows how far and in what direction this particular height deviates from the average.

Demystifying Statistical Analysis

Statistical analysis is a field dedicated to collecting, exploring, and presenting large amounts of data to uncover patterns and trends. This process includes using descriptive statistics, such as mean and standard deviation, to summarize data.

• Descriptive statistics help in understanding the main characteristics of a dataset.
• Inferential statistics allow for predictions or conclusions about a population based on a sample.

The exercise on Z-scores is a component of statistical analysis that shows how the height of 165 cm can be interpreted statistically. By understanding that this height is half a standard deviation below the mean, one can make informed conclusions about how normal or unusual this measurement is within the given distribution.