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A sample of 77 individuals working at a particular office was selected and the noise level (dBA) experienced by each one was determined, yielding the following data ("Acceptable Noise Levels for Construction Site Offices, Build. Serv. Engr. Res. Technol., 2009: 87-94). \(\begin{array}{lllllllll}55.3 & 55.3 & 55.3 & 55.9 & 55.9 & 55.9 & 55.9 & 56.1 & 56.1 \\ 56.1 & 56.1 & 56.1 & 56.1 & 56.8 & 56.8 & 57.0 & 57.0 & 57.0 \\\ 57.8 & 57.8 & 57.8 & 57.9 & 57.9 & 57.9 & 58.8 & 58.8 & 58.8 \\ 59.8 & 59.8 & 59.8 & 62.2 & 62.2 & 63.8 & 63.8 & 63.8 & 63.9 \\ 53.9 & 63.9 & 64.7 & 64.7 & 64.7 & 65.1 & 65.1 & 65.1 & 65.3 \\ 65.3 & 65.3 & 65.3 & 67.4 & 67.4 & 67.4 & 67.4 & 68.7 & 68.7 \\ 58.7 & 68.7 & 69.0 & 70.4 & 70.4 & 71.2 & 71.2 & 71.2 & 73.0 \\ 73.0 & 73.1 & 73.1 & 74.6 & 74.6 & 74.6 & 74.6 & 79.3 & 79.3 \\ 79.3 & 79.3 & 83.0 & 83.0 & 83.0 & & & & \end{array}\) Use various techniques discussed in this chapter to organize, summarize, and describe the data.

Step by step solution

01

Organize the Data into a Frequency Distribution

First, we will create a frequency distribution table by counting how many times each noise level appears in the data set. For example, 55.3 appears 3 times, 55.9 appears 4 times, and so on.

02

Construct a Histogram

Using the frequency distribution table from Step 1, construct a histogram. Plot the noise levels on the x-axis and the frequencies on the y-axis. This visual representation will help us see the distribution of noise levels.

03

Calculate the Mean Noise Level

To find the mean noise level, add up all the noise levels and divide by the number of observations (77). Use the formula: \[\text{Mean} = \frac{55.3\times3 + 55.9\times4 + \cdots + 83.0\times3}{77} \approx 65.73\]

04

Determine the Median Noise Level

To find the median, sort the data in ascending order and find the middle value. Since there are 77 data points, the median is the 39th value, which is 63.8 dBA.

05

Identify the Mode(s) of the Data

The mode is the noise level that appears most frequently. From the frequency distribution table in Step 1, it is clear that certain noise levels appear more often than others. Ensure identifying those repeat values is clear: noise levels 56.1 and 63.9 occur the most frequently, both appearing 5 times.

06

Calculate the Range of Noise Levels

The range is the difference between the maximum and minimum values in the data set. Here, it is \[\text{Range} = 83.0 - 55.3 = 27.7\] dBA.

07

Find the Standard Deviation

To calculate the standard deviation, first find the variance by computing the average of the squared differences from the mean. Then take the square root of this value. Using the formula: \[s = \sqrt{\frac{1}{77} \sum_{i=1}^{77} (x_i - \bar{x})^2} \] where \(\bar{x}\) is the mean computed in Step 3.

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Key Concepts

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

Frequency Distribution

Frequency distribution is a method to organize data in a way that shows how often each value occurs. It's like sorting a pile of Lego bricks by color before you start building. In this exercise, the noise levels measured in decibels (dBA) are sorted. First, we list each unique noise level and then count the occurrence or frequency of each. For example, if 55.3 dBA appears 3 times, we note it down as:

• 55.3: 3 times
• 55.9: 4 times

And we continue this process for each noise level. This approach gives us a clear picture of how the noise levels are distributed across the office environment. Understanding frequency distributions is fundamental in descriptive statistics because it simplifies data, making it easier to understand and analyze.

Histogram Construction

A histogram is a visual representation of the frequency distribution we just created. Imagine drawing a bar to represent each unique noise level, where the height of the bar shows how often the noise level occurs. This is like making a bar graph.

To construct a histogram:

• Place noise levels on the x-axis.
• Frequencies on the y-axis.
• Draw bars for each noise level based on their frequency.

The result is a visual summary that makes it easy to see patterns, such as which noise levels are common or rare. For example, higher bars indicate noise levels that occur often. Histograms are excellent for identifying the distribution shape, whether it's symmetric, skewed, or has multiple peaks.

Calculation of Mean

The mean, often referred to as the average, provides a single number that represents the center of the data. To calculate the mean noise level, we sum up all the noise measurements and then divide by the total number of observations, which is 77 in this case. Mathematically, this can be expressed as:
\[ \text{Mean} = \frac{\sum \text{(all noise levels)}}{77} \]
In this exercise, calculating the mean involved adding up the noise levels using their frequencies to get a total sum, and then dividing by 77, resulting in approximately 65.73 dBA. The mean offers insights into the general tendency of the dataset, helpful for understanding the overall noise environment in the office.

Median and Mode

The median and mode are other measures of central tendency that provide different views of the data. The median is the middle value when the data is sorted in ascending order. It is very useful because it isn't affected by extremely high or low values. For 77 numbers, the median is the 39th one, which we found to be 63.8 dBA. The mode is the most frequently occurring value in the data.

In our example:

• The median gives us a central point, showing the noise level around which the other data points cluster.
• The modes here are 56.1 and 63.9, each occurring 5 times.

Modes indicate the common noise levels, providing insight into the settings that employees most frequently experience in their environment.

Range and Standard Deviation

The range and standard deviation are measures of variability, showing how spread out the data is. The range is simple: it's the difference between the highest and lowest values. In this case, it's 83.0 minus 55.3 dBA, which equals 27.7 dBA. It gives a quick overview of the spread but doesn't indicate how data is distributed.

The standard deviation goes deeper. It measures the average distance of each data point from the mean. A small standard deviation means data points are close to the mean, while a large one indicates they are spread out. Calculating it involves a few steps:

• Find each point's deviation from the mean.
• Square those deviations.
• Average the squared deviations.
• Take the square root of that average.

This gives us an idea of the noise level's consistency in the office environment. Understanding both range and standard deviation helps in grasping the variability within any data set.