Problem 71 - Basic Statistics for Business and Economics with Student

Question

A southern state has seven state universities in its system. The numbers of volumes (in thousands) held in its libraries are \(83,510,33,256,401,47,\) and \(23 .\) a. Is this a sample or a population? b. Compute the standard deviation.

Step-by-Step Solution

Answer

a. This is a population. b. The standard deviation is approximately 179.93.

Step 1: Determine if it's a Sample or Population

Since the data includes all state universities within the system, this represents the entire group that we're interested in. Therefore, this is a population and not a sample.

Step 2: Find the Mean

Calculate the mean (average) number of volumes by summing all the numbers and dividing by the total count of these numbers.Sum: \(83 + 510 + 33 + 256 + 401 + 47 + 23 = 1353\)Mean: \(\frac{1353}{7} = 193.29\)

Step 3: Calculate Variance for Population

Find the variance by computing the squared difference of each data value from the mean, summing them, and dividing by the number of data points.Variance Formula for Population: \[ \sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2 \]\((83 - 193.29)^2 = 12148.16\)\((510 - 193.29)^2 = 100264.94\)\((33 - 193.29)^2 = 25645.16\)\((256 - 193.29)^2 = 3929.44\)\((401 - 193.29)^2 = 43092.64\)\((47 - 193.29)^2 = 21407.84\)\((23 - 193.29)^2 = 29085.44\)Sum of squared differences: \(226573.62\)Variance: \(\frac{226573.62}{7} = 32367.66\)

Step 4: Compute the Standard Deviation

The standard deviation is the square root of the variance. For our data, this will be \(\sqrt{32367.66} \approx 179.93\)

Key Concepts

Population vs. Sample

When discussing statistics, it's essential to determine whether we are working with a sample or a population. The difference lies in the extent of the data we have:

• Population: This includes all members of a specific group or category. For instance, if we consider every state university within a particular system, this collection of data is a population.
• Sample: This is a subset of the population and may not cover every member. We often use samples when dealing with larger populations.

In the original exercise, the numbers of volumes in libraries at all the state's universities were included, making it a population study. Recognizing whether you're working with a population or a sample is crucial, as it affects the formulas used in calculations, such as variance and standard deviation.

Variance Calculation

Variance helps us understand how spread out our data is by measuring the average squared deviation from the mean. To calculate variance, follow these steps:

• Compute the mean (average) of your data set.
• Subtract the mean from each data point to find the deviation of each value.
• Square these deviations to eliminate negative values.
• For a population, divide the sum of squared deviations by the number of data points.
• For a sample, divide by one less than the number of data points (degrees of freedom).

Using the exercise data as an example, we calculated the variance using the population formula:\[ \sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2 \]Where \(N\) is the total number of volumes, yielding a variance of 32367.66.

Mean Calculation

The mean or average provides a central value for a set of numbers and is a commonly used measure of central tendency.To find the mean:

• Add all numbers in the data set together to find the total sum.
• Divide this sum by the count of numbers you have.

For the library volume data:
1. Total sum of volumes is computed as:\[83 + 510 + 33 + 256 + 401 + 47 + 23 = 1353\]2. Count of volumes is 7 (since there are 7 universities).3. Mean calculation:\[\mu = \frac{1353}{7} \approx 193.29\]This mean shows us an average number of volumes held across these universities' libraries. Calculating the mean is an initial and essential step before moving onto variance and standard deviation, as it provides the baseline for measuring deviations.