Problem 27 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The number of average hours worked per year per worker in the United States and five European countries in 2002 is given in the following table: $$\begin{array}{lcccccc}\hline & & & \text { Great } & & \text { West } & \\\\\text { Country } & \text { U.S. } & \text { Spain } & \text { Britain } & \text { France } & \text { Germany } & \text {Norway } \\ \hline \text { Average } & & & & & & \\\\\text { Hours } & 1815 & 1807 & 1707 & 1545 & 1428 & 1342 \\ \text { Werked } & & & & & & \\ \hline\end{array}$$ Find the average of the average hours worked per worker in 2002 for workers in the six countries. What is the standard deviation for these data?

Step-by-Step Solution

Answer

The average hours worked per worker in the six countries in 2002 is 1607 hours, with a standard deviation of approximately 181.90 hours.

Step 1: Calculate the mean

To find the mean (average) hours worked in 2002 for workers in the six countries, we will add up the hours worked in each country and divide the sum by the number of countries (6). Mean (average) = \(\frac{\text{Sum of hours worked}}{\text{Number of countries}}\) The sum of hours worked in each country is: \(1815 + 1807 + 1707 + 1545 + 1428 + 1342 = 9644\) Mean (average) = \(\frac{9644}{6} = 1607\) The mean or average hours worked per worker in the six countries in 2002 is 1607 hours.

Step 2: Calculate the squared difference between each data point and the mean

Now we need to calculate the squared differences between each data point and the mean. The formula for this is: Squared difference = \((\text{data point} - \text{mean})^2\) For each country, we will find the squared difference and note it down. - US: \((1815 - 1607)^2 = 43264\) - Spain: \((1807 - 1607)^2 = 40000\) - Great Britain: \((1707 - 1607)^2 = 10000\) - France: \((1545 - 1607)^2 = 3844\) - West Germany: \((1428 - 1607)^2 = 32041\) - Norway: \((1342 - 1607)^2 = 70369\)

Step 3: Calculate the variance

Now, we will calculate the variance of the data set. The variance is the average of the squared differences. Variance = \(\frac{\text{Sum of squared differences}}{\text{Number of countries}}\) The sum of the squared differences is: \(43264 + 40000 + 10000 + 3844 + 32041 + 70369 = 198518\) Variance = \(\frac{198518}{6} = 33086.333\)

Step 4: Calculate the standard deviation

Finally, to find the standard deviation, we will take the square root of the variance. Standard deviation = \(\sqrt{\text{Variance}}\) Standard deviation = \(\sqrt{33086.333} = 181.90\) The standard deviation of the average hours worked per worker in the six countries in 2002 is approximately 181.90 hours.

Key Concepts

Mean Calculation

In statistics, the mean is a central value that is often referred to as the "average." It’s a straightforward way to measure the central tendency of a given data set. Calculating the mean is simple: you add up all the numbers in the data set and then divide the total by the number of data points. This gives you a single number that represents an average of the data.
For example, let’s consider the exercise where we have the number of hours worked per year in six different countries. By summing up these values: 1815, 1807, 1707, 1545, 1428, and 1342, we get a total of 9644. Since there are six countries, the mean is calculated by dividing 9644 by 6. Therefore, the mean average hours worked is 1607 hours.
This mean value helps in understanding what an average worker’s hours might look like across these countries, smoothing out variations and extremes in the data.

Variance and Standard Deviation

While the mean offers a key insight into the general trend of data, it alone doesn’t convey how spread out the data points are from the mean itself. This is where variance and standard deviation come into play.
The variance is essentially the average of the squared differences from the mean. You first determine how far each data point is from the mean, square that number (to remove any negative signs and emphasize larger deviations), and then average these squared differences. From the exercise:

• US: (1815 - 1607)^2 = 43264
• Spain: (1807 - 1607)^2 = 40000
• Great Britain: (1707 - 1607)^2 = 10000
• France: (1545 - 1607)^2 = 3844
• Germany: (1428 - 1607)^2 = 32041
• Norway: (1342 - 1607)^2 = 70369

This step helps capture the degree of spread in the data. Adding up these squared differences gives us 198518, and dividing by six provides a variance of 33086.333.
The standard deviation, on the other hand, is merely the square root of the variance. It helps revert the variance back to the original units of measure, making it more interpretable. In this case, the standard deviation is approximately 181.90 hours, indicating how much the average work hours vary from the mean for these countries.

Data Analysis

Data analysis involves examining data sets to draw conclusions about the information they contain with the help of various statistical tools. In this exercise, by computing the mean, variance, and standard deviation, we can interpret the work hour data better.
Understanding the mean provides a general overview, while variance and standard deviation offer insight into data spread and variation. This analysis helps us see that although there is a significant mean of 1607 hours, the data is spread across a broad spectrum based on the standard deviation. This suggests variability in working conditions or practices across those countries.
Data analysis like this is crucial not only in education but in understanding socio-economic patterns. Such statistical interpretations can guide policymakers and educators in decision-making by highlighting areas for improvement or change. It lays the foundation for more in-depth research and offers a clear picture of the current situation.