Problem 10 - Algebra 2 and Trigonometry

Question

A set of data is normally distributed with a mean of 40 and a standard deviation of \(5 .\) Find a data value that is: a. 1 standard deviation above the mean b. 2.4 standard deviations above the mean c. 1 standard deviation below the mean d. 2.4 standard deviations below the mean

Step-by-Step Solution

Answer

a. 45, b. 52, c. 35, d. 28

Step 1: Understand the Problem

We need to find specific data values in a normally distributed dataset where the mean is 40 and the standard deviation is 5. Our task is to determine the data values for certain standard deviations above and below the mean.

Step 2: Calculate 1 Standard Deviation Above the Mean

A value that is 1 standard deviation above the mean is calculated by adding 1 times the standard deviation to the mean:\[Mean + 1 imes ext{Standard Deviation} = 40 + 1 imes 5 = 45\]Thus, the data value is 45.

Step 3: Calculate 2.4 Standard Deviations Above the Mean

A value that is 2.4 standard deviations above the mean is found by adding 2.4 times the standard deviation to the mean:\[Mean + 2.4 imes ext{Standard Deviation} = 40 + 2.4 imes 5 = 52\]So, the data value is 52.

Step 4: Calculate 1 Standard Deviation Below the Mean

A value that is 1 standard deviation below the mean is calculated by subtracting 1 times the standard deviation from the mean:\[Mean - 1 imes ext{Standard Deviation} = 40 - 1 imes 5 = 35\]The data value is 35.

Step 5: Calculate 2.4 Standard Deviations Below the Mean

A value that is 2.4 standard deviations below the mean is calculated by subtracting 2.4 times the standard deviation from the mean:\[Mean - 2.4 imes ext{Standard Deviation} = 40 - 2.4 imes 5 = 28\]Hence, the data value is 28.

Key Concepts

Understanding the Mean

The mean is a central concept in statistics, often referred to as the 'average.' It represents a central point or typical value within a dataset. Consider a normal distribution curve, which is symmetrical. The mean is located right at the center of this curve.

For the dataset in our example, the mean is 40.
Determining this value helps in understanding the center point of the data and provides a baseline or reference for measuring how other data points are dispersed or clustered around it. The mean acts as an anchor in a normal distribution, allowing us to measure variations easily with concepts like standard deviation and deviations from the mean.

Exploring Standard Deviation

Standard deviation is a measure of the dispersion or spread of data values around the mean. This term quantifies how data values vary from the mean, revealing the extent of diversity in a dataset.

In our exercise, the standard deviation is provided as 5.

• If the standard deviation is small, data points are closely packed near the mean.
• If the standard deviation is large, data values spread out over a wider range.

In a normal distribution, approximately 68% of data values fall within one standard deviation from the mean, 95% fall within two standard deviations, and 99.7% fall within three standard deviations. Using standard deviation alongside the mean helps locate data values and analyze the distribution more precisely.

Significance of Data Values

Data values in a distribution tell us where specific observations lie concerning the mean. Given our mean of 40 and standard deviation of 5:

• A data value 1 standard deviation above the mean is 45.
• A data value 2.4 standard deviations above the mean is 52.
• A data value 1 standard deviation below the mean is 35.
• A data value 2.4 standard deviations below the mean is 28.

These calculations are crucial for identifying positions of data within the distribution. They also indicate frequency and likelihood of those observations in the dataset.

Calculating Above and Below the Mean

Understanding how to find data values above and below the mean requires you to manipulate the mean by a multiple of the standard deviation. This process helps determine how far a data point belongs from the mean.

To find these values, you adjust the mean by either adding or subtracting the number of standard deviations specified:

• For values above the mean, add the product of the standard deviation and the number of standard deviations.
• For values below, subtract it.

This straightforward mathematical operation reveals not only the position of data points but also delivers insights into their dispersion along the normal distribution curve.