Problem 60 - Thinking Mathematically

Question

A set of data items is normally distributed with a mean of 400 and a standard deviation of 50. In Exercises \(59-66\), find the data item in this distribution that corresponds to the given z-score. \(z=3\)

Step-by-Step Solution

Answer

The data item in the distribution that corresponds to the \(z=3\) is \(550\). Remember that this means the data item is 3 standard deviations above the mean.

Step 1: Identify the mean (μ) and standard deviation (σ)

The mean μ of the normally distributed data is given as 400, and the standard deviation σ is given as 50.

Step 2: Identify the z-score

The Z score for the task at hand is given as 3.

Step 3: Convert the z score to an x score

To find the X score that corresponds to the given Z score, use the formula \(X = μ + zσ \). Substituting the given values into the formula, we get \(X = 400 + 3*50 = 400 + 150 = 550\)

Key Concepts

Understanding the Z-Score

The z-score is a statistical measurement that describes a value's relation to the mean of a group of values. It is expressed as the number of standard deviations away from the mean a specific data point lies. This score helps to understand how far away a data point is from the center of a normal distribution.

The formula to find the z-score is:

• \[ Z = \frac{(X - μ)}{σ} \]

Here, \( X \) is the value in the dataset, \( μ \) is the mean, and \( σ \) is the standard deviation.

A positive z-score indicates a value above the mean, while a negative z-score shows it is below. In our example, given that \( z = 3 \), it tells us that the data value is 3 standard deviations above the mean.

What is the Mean?

The mean, often called the average, is a measure of central tendency that sums all values in a dataset and divides by the number of values. In simpler terms, it gives you an idea of the typical value within the dataset.

To calculate the mean:

• Sum all the values in the dataset
• Divide by the number of values in the dataset

For example, if the dataset is \[ [10, 20, 30, 40, 50] \], the mean would be \( \frac{150}{5} = 30 \).

In our problem, the mean \( (μ) \) is given as 400, meaning this is the average value around which all other data values in the dataset are distributed.

Role of Standard Deviation

Standard deviation is a measure of how spread out the numbers in a dataset are. More technically, it calculates the extent of deviation for a group of data points from their mean or average value.

To calculate standard deviation:

• Find the difference between each data point and the mean
• Square these differences
• Find the average of these squared differences
• Take the square root of this average

The smaller the standard deviation, the closer the data values are to the mean. Conversely, a larger standard deviation means a wider spread.

In the original exercise, the standard deviation \( (σ) \) is 50, meaning on average, each data point is 50 units away from the mean.

Exploring the Data Set

A data set is simply a collection of data points. In statistical analysis, data sets are often assumed to have a certain distribution, such as normal distribution, which is symmetric around the mean.

Normal distributions are characterized by the bell-shaped curve and are essential because they closely approximate many natural phenomena. In our example, knowing the data follows a normal distribution makes it possible to use concepts like the z-score.

With a given mean and standard deviation, we can determine where most values fall in the data set. The accommodation of z-scores in normal distributions allows identification of the position of a specific data point, like what the exercise requires with \( z=3 \). This showcases the utility of these tools in analyzing datasets.

Mathematical Solution Made Easy

Solving mathematical problems using statistics doesn't have to be complicated. Knowing the formula and how to apply it to given conditions makes obtaining the solution straightforward.

In the given exercise, we calculated the data item that corresponds to a z-score of 3. Using the formula:

• \[ X = μ + zσ \]

Where \( μ = 400 \), \( z = 3 \), and \( σ = 50 \), the steps were:

• Multiply the z-score by the standard deviation: \( 3 \times 50 = 150 \)
• Add this product to the mean: \( 400 + 150 = 550 \)

Hence, the data point we're interested in is 550. This approach underlines the importance of having a strong grasp of the basic concepts to solve problems efficiently.