Problem 27 - Algebra 2

Question

In a set of data, the value that is \(-3\) standard deviations from the mean is \(86 .\) The value that is \(+1\) standard deviation from the mean is \(250 .\) a. Find the mean. b. Find the standard deviation. c. Find an interval with an end value of 250 that contains about 81.5\(\%\) of the data.

Step-by-Step Solution

Answer

The mean of the data set is 209 and the standard deviation is 41. The interval containing about 81.5% of the data has end numbers at 168 and 250.

Step 1: Find the Mean

We know that the formula representing deviation of a value from the mean (represented by X), is: \(X = \mu + Z\sigma\), where \(\mu\) is the mean, Z is the number of standard deviations, and \(\sigma\) is the standard deviation. We apply this formula twice using the values given in the problem for when Z = -3 and Z = 1. Therefore, we have the following two equations: \[86 = \mu - 3\sigma\] \[250 = \mu + \sigma\]. Subtracting the first equation from the second will allow us to solve for the standard deviation, \(\sigma\).

Step 2: Find the Standard Deviation

By subtracting the first equation from the second, we get: \(164 = 4\sigma\). Solving for \(\sigma\), we find that the standard deviation is \(41\). Then, substitute \(\sigma = 41\) into one of the original equations, say \[250 = \mu + 41\], to solve for \(\mu\), the mean.

Step 3: Find the Interval Containing 81.5% of the data

Knowing that 81.5% of the data lies within Z = -1 and Z = 1, we can find the interval by substituting these values and our calculated mean and standard deviation into the formula \(X = \mu + Z\sigma\): When Z = -1, \(X = 209 - 41 = 168\). When Z = 1, \(X = 209 + 41 = 250\). Therefore, the interval containing 81.5% of the data has end numbers 168 and 250.

Key Concepts

Understanding the Mean

The mean is a central value in a data set. It is often called the "average," as it represents the typical value. You calculate the mean by adding up all the values and then dividing by the number of values. In our exercise, we determined the mean using the formula from the deviation of each data point: \(86 = \mu - 3\sigma\) and \(250 = \mu + \sigma\). By solving these equations, we found that the mean \(\mu\) is the balance point from which deviations are measured. In simpler terms, it tells us where the middle of our data is.

Exploring the Interval

An interval in statistics indicates a range of values. It helps understand where most of the data points lie. In this exercise, we need an interval that contains 81.5% of the data. Using the standard deviation and the mean, the interval from \(168\) to \(250\) captures this percentage.

• The standard deviation tells us how spread out the data is.
• For a normal distribution, specific percentages of data fall within certain standard deviations from the mean.
• The values at \(Z = -1\) and \(Z = 1\) help identify this crucial interval.

Understanding this interval is vital for analyzing how the data is distributed.

What is a Data Set?

A data set is simply a collection of values or observations. These can be numbers, words, or symbols. In statistics, data sets are analyzed to find patterns or make decisions. Our exercise examines specific numbers within a data set that relates to the mean and standard deviation.

• Data sets can be small or large depending on what's being studied.
• The values can be computed or observed from experiments or surveys.
• Analyzing a data set often involves finding the mean, median, mode, and range.

Getting familiar with data sets is crucial for comprehending real-world information.

Z-score Simplified

A Z-score is a measure that describes a value's relation to the mean of a data set. It shows how many standard deviations a specific data point is from the mean. In our exercise, Z-scores of \(-3\) and \(1\) were used to find specific values in the data set."The Z-score formula is: \(Z = \frac{X - \mu}{\sigma}\)." It helps identify how unusual or typical a value is within a data set.

• Positive Z-scores indicate values above the mean.
• Negative Z-scores indicate values below the mean.
• Z-scores are essential in standard normal distributions where mean is \(0\) and standard deviation is \(1\).

Understanding Z-scores helps make sense of how data deviates from the expected norm.