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How much time does it take to travel to work? Table 1.36 shows the mean commute time by state for workers at least 16 years old who are not working at home. Find the mean travel time, and round off the answer properly. $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline 24.0 & {24.3} & {25.9} & {18.9} & {27.5} & {17.9} & {21.8} & {20.9} & {16.7} & {27.3} \\ \hline 18.2 & {24.7} & {20.0} & {22.6} & {23.9} & {18.0} & {31.4} & {22.3} & {24.0} & {25.5} \\\ \hline 24.7 & {24.6} & {28.1} & {24.9} & {22.6} & {23.6} & {23.4} & {25.7} & {24.8} & {25.5} \\ \hline 21.2 & {25.7} & {23.1} & {23.0} & {23.9} & {26.0} & {16.3} & {23.1} & {21.4} & {21.5} \\ \hline 27.0 & {27.0} & {31.6} & {31.7} & {23.3} & {30.1} & {22.9} & {23.3} & {21.7} & {18.6} \\ \hline \end{array}$$ Table 1.36

Step by step solution

01

Calculate the Total Sum

Add all the mean travel times provided in Table 1.36. You will be summing up these 50 numbers: 24.0, 24.3, 25.9, 18.9, 27.5, 17.9, 21.8, 20.9, 16.7, 27.3, 18.2, 24.7, 20.0, 22.6, 23.9, 18.0, 31.4, 22.3, 24.0, 25.5, 24.7, 24.6, 28.1, 24.9, 22.6, 23.6, 23.4, 25.7, 24.8, 25.5, 21.2, 25.7, 23.1, 23.0, 23.9, 26.0, 16.3, 23.1, 21.4, 21.5, 27.0, 27.0, 31.6, 31.7, 23.3, 30.1, 22.9, 23.3, 21.7, and 18.6.

02

Compute the Mean

Divide the total sum calculated in Step 1 by the total number of states, which is 50. This calculation will give you the mean travel time.

03

Round the Mean

Round the mean calculated in Step 2 to one decimal place, as requested by the exercise.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arithmetic Mean

Calculating the arithmetic mean is a basic yet vital statistical operation. It helps us understand the average or central value in a data set. To find the arithmetic mean, follow these steps:

• First, add together all the numbers you have in your data set. This is called summation. For instance, if you have a list of times, you add all the times together.
• Next, divide the total sum you calculated by the quantity of numbers in your data set. This gives you the average.

In this example from Table 1.36, there are 50 data points representing the mean commute time in different states. The arithmetic mean provides a simple summary measure of these times by putting all the values together and then dividing by 50. As a result, you find out how long, on average, it takes workers across different states to commute. This calculation makes it much easier to grasp the general trend in the data.

Data Summation

Data summation involves adding up all the elements in a data set to obtain a total sum, which is the first step in calculating the arithmetic mean. The actual sum provides critical insight into the data's collective value.

In our commute time example, you add numbers like 24.0, 24.3, 25.9, and so on until you have included all 50 figures. It's essential to be careful and methodical to ensure accuracy with such large data sets. Using a calculator can help minimize errors.

Summation allows us to compress a broad set of numbers into a single, more manageable figure. This total can be further used to derive other statistical measures or simply give us an overview of the data set's magnitude.

Rounding Numbers

Rounding is an essential skill in mathematics, enabling us to simplify numbers for ease of understanding or reporting. After computing the arithmetic mean, we often round it to make it more intuitive.

To round to one decimal place:

• Identify the number in the tenths place, which is the first digit to the right of the decimal point.
• Look at the number immediately following it (hundredths place). If this number is 5 or greater, increase the tenths digit by one. If it is less than 5, keep the tenths digit unchanged.

In the case of our mean commute time example, we round to one decimal place as instructed. This results in a cleaner, more digestible figure that maintains a reasonable balance between simplicity and precision. Understanding when and how to round numbers effectively is key in various practical situations, ensuring the data presented is both accurate and accessible.