Problem 14 - Algebra 2 and Trigonometry

Question

In \(11-14,\) select the numeral that precedes the choice that best completes the statement or answers the question. The heights of 200 women are normally distributed. The mean height is 170 centimeters with a standard deviation of 10 centimeters. What is the best estimate of the number of women in this group who are between 160 and 170 centimeters tall? $$ \begin{array}{llll}{\text { (1) } 20} & {\text { (2) } 34} & {\text { (3) } 68} & {\text { (4) } 136}\end{array} $$

Step-by-Step Solution

Answer

The best estimate is 68 women.

Step 1: Understanding the Normal Distribution

For a normal distribution, the mean (\(\mu\)) is the center of the distribution, and the standard deviation (\(\sigma\)) measures how spread out the values are. In our case, \(\mu = 170\) cm and \(\sigma = 10\) cm.

Step 2: Convert Heights to Z-Scores

To compare data using the normal distribution table, we convert heights to Z-scores. The Z-score formula is \( Z = \frac{X - \mu}{\sigma} \). For 160 cm: \( Z = \frac{160 - 170}{10} = -1 \), and for 170 cm: \( Z = \frac{170 - 170}{10} = 0 \).

Step 3: Find Probabilities from Z-Table

Look up the probabilities corresponding to the Z-scores in a standard normal distribution table. The probability for \( Z = 0 \) is 0.5000, and for \( Z = -1 \) is about 0.1587. This means the probability of a woman being between 160 cm and 170 cm is \(0.5000 - 0.1587 = 0.3413\) or 34.13%.

Step 4: Calculate Number of Women

Multiply the probability of a woman being between 160 cm and 170 cm by the total number of women. \(0.3413 \times 200 = 68.26\). Since we cannot have a fraction of a woman, we round to the nearest whole number, which is 68.

Key Concepts

Understanding the Z-Score

The Z-score is a concept in statistics that allows you to determine how far a data point is from the mean of a data set. It is calculated by subtracting the mean from the data point and then dividing by the standard deviation. In mathematical terms, the formula is given by \[ Z = \frac{X - \mu}{\sigma} \]where:

• \( X \) is the data point
• \( \mu \) is the mean
• \( \sigma \) is the standard deviation

A Z-score tells you how many standard deviations away from the mean your data point is located. For example, a Z-score of 0 indicates that the data point is exactly at the mean. A positive Z-score indicates the data point is above the mean, while a negative one indicates it is below. In the context of a normal distribution, Z-scores are useful because they allow us to compare data points that come from different normal distributions, transforming them into a standard form. This makes it possible to use the standard normal distribution table (or Z-table) to find probabilities associated with different data points.

Delving into Standard Deviation

Standard deviation is a statistical measurement that shows the amount of variation or dispersion in a set of values. It measures how much the individual numbers in a data set deviate from the mean. The formula for standard deviation is:\[ \sigma = \sqrt{\frac{1}{N}\sum_{i=1}^N (X_i - \mu)^2} \]where:

• \( \sigma \) denotes the standard deviation
• \( N \) is the number of observations
• \( X_i \) represents each value in the data set
• \( \mu \) is the mean of the data

A low standard deviation means that most of the numbers are close to the mean, whereas a high standard deviation means the numbers are more spread out. In a normal distribution, about 68% of the values lie within one standard deviation of the mean. Thus, understanding standard deviation helps us interpret how concentrated or spread out the data values are relative to the mean. This plays a key role in various statistical analyses.

The Significance of Mean in a Data Set

The mean, often referred to as the average, is one of the most common measures of central tendency in statistics. It represents the sum of all the values in a data set divided by the number of values. The formula for the mean \( \mu \) is: \[ \mu = \frac{\sum_{i=1}^N X_i}{N} \]where:

• \( N \) is the total number of values
• \( \sum_{i=1}^N X_i \) is the sum of all values in the data set

The mean provides a central value around which the data points tend to cluster. In the context of a normal distribution, the mean is the peak of the bell-shaped curve, indicating where most values in the dataset occur. Its simplicity makes it particularly useful in describing large data sets. However, it can be affected by extreme values or outliers. In such distributions, the mean may not accurately reflect the 'center' of the data. Understanding the mean helps provide context for other statistical measures, such as variance and standard deviation.