Problem 13 - 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 10 -year-old children are normally distributed with a mean of 138 centimeters with a standard deviation of 5 centimeters. The height of a 10 -year-old child who is as tall as or taller than 95.6\(\%\) of all 10 -year-old children is $$ \begin{array}{ll}{\text { (1) between } 138 \text { and } 140 \mathrm{cm} .} & {\text { (2) between } 140 \text { and } 145 \mathrm{cm} .} \\ {\text { (3) between } 145 \text { and } 148 \mathrm{cm} .} & {\text { (4) taller than } 148 \mathrm{cm} .}\end{array} $$

Step-by-Step Solution

Answer

(3) between 145 and 148 cm.

Step 1: Identify the Problem

We need to find the height that corresponds to the 95.6\(\%\) percentile of a normally distributed variable, where the mean is 138 cm and the standard deviation is 5 cm.

Step 2: Understand the Normal Distribution Parameters

The problem states that the mean (\(\mu\)) is 138 cm and the standard deviation (\(\sigma\)) is 5 cm. This information helps us understand the spread and center of the distribution.

Step 3: Convert the Percentile to a Z-Score

The 95.6\(\%\) percentile corresponds to a Z-score. Using a Z-table or calculator, we find that the approximate Z-score for 95.6\(\%\) is 1.67.

Step 4: Apply the Z-Score Formula

The Z-score formula is \(Z = \frac{X - \mu}{\sigma}\). We rearrange it to find \(X\): \(X = Z \cdot \sigma + \mu\). Substitute the values: \(X = 1.67 \cdot 5 + 138\).

Step 5: Calculate the Height

Perform the calculation: \(X = 1.67 \times 5 + 138 = 8.35 + 138 = 146.35\). This height is rounded to 146 cm.

Step 6: Select the Correct Range

The calculated height 146 cm falls in the range 145-148 cm. Hence, the correct option is (3) between 145 and 148 cm.

Key Concepts

Percentile

A percentile is a measure used in statistics to indicate the value below which a given percentage of observations fall. For example, if you are at the 95th percentile for height, it means you are taller than 95% of your peers in the data set. Percentiles are useful in comparing different data sets or understanding an individual's position within a specific distribution. Percentiles divide the data into 100 equal parts, with each part representing 1% of the data. In the context of a normal distribution, percentiles help in identifying boundaries of data points. For example, a 95th percentile point tells us where the top 5% of the data lies. This is crucial for making predictions and decisions based on statistical data.

Z-Score

A Z-score is a numerical measurement that describes a value's relationship to the mean of a group of values. It tells us how many standard deviations a particular point is from the mean.To compute a Z-score, we use the formula:\[ Z = \frac{X - \mu}{\sigma} \]Where:

• \(Z\) is the Z-score,
• \(X\) is the value for which you want to calculate the Z-score,
• \(\mu\) is the mean of the population,
• \(\sigma\) is the standard deviation of the population.

Positive Z-scores indicate values above the mean, while negative Z-scores show values below the mean. By converting a percentile into a Z-score, we can easily determine how far a particular data point is from the average, helping us make sense of where it stands in the normal distribution.

Mean and Standard Deviation

The mean and standard deviation are key statistical measures that characterize a normal distribution. The mean, often symbolized by \(\mu\), represents the average of the data set. It is the central value around which the data is distributed. The standard deviation, denoted by \(\sigma\), measures 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, whereas a high standard deviation means the values are spread out over a larger range. These two parameters are foundational in understanding the spread and behavior of data in a normal distribution. They allow us to calculate Z-scores and to predict the probabilities of observing specific data points within the population.

Normal Distribution Table

A normal distribution table, also known as a Z-table, is a mathematical table used to find the percentage that corresponds to a particular Z-score on the standard normal distribution. It provides the cumulative probability associated with each Z-score, showing how likely a certain value is to occur within the normal distribution. The table typically shows probabilities for Z-scores ranging from negative to positive values. It is widely used to convert between Z-scores and percentiles, enabling accurate predictions and statistical analysis. Using a Z-table is straightforward — you simply locate the Z-score of interest along the rows and columns. The intersection gives you the area under the curve up to that point as a percentage of the total distribution. This helps in determining the likelihood of observing a value or in converting a categorical percentile into a numerical measurement useful for statistical analysis.