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Chapter 27: Problem 3

The following are the values obtained by a class of 14 students when measuring a physical quantity \(x: 53.8,53.1,56.9,54.7,58.2,54.1,56.4,54.8,57.3,51.0,55.1\) \(55.0,54.2,56.6\) (a) Display these results as a histogram and state what you would give as the best value for \(x\). (b) Without calculation estimate how much reliance could be placed upon your answer to (a). (c) Databooks give the value of \(x\) as \(53.6\) with negligible error. Are the data obtained by the students in conflict with this?

Short Answer

Expert verified
Best value for x is approximately 54.5. Reliability is moderate. Data is not in conflict with databook value of 53.6.

Step by step solution

01

Organize the Data

List all the values given: 53.8, 53.1, 56.9, 54.7, 58.2, 54.1, 56.4, 54.8, 57.3, 51.0, 55.1, 55.0, 54.2, 56.6.
02

Create Bins for Histogram

Decide the range for each bin, for example: 50-51, 51-52, 52-53, 53-54, 54-55, 55-56, 56-57, 57-58, 58-59.
03

Count the Values in Each Bin

Count how many values fall into each bin: - Bin 50-51: 1 - Bin 51-52: 1 - Bin 52-53: 0 - Bin 53-54: 2 - Bin 54-55: 5 - Bin 55-56: 2 - Bin 56-57: 3 - Bin 57-58: 1 - Bin 58-59: 1
04

Draw the Histogram

Using the bin counts, draw the histogram with bins on the x-axis and frequency on the y-axis.
05

Determine the Best Value for x

Identify the bin with the highest frequency, which is 54-55. The best value for x is within this range, so the best estimate of x would be around 54.5.
06

Estimate the Reliability

Since the values are spread out over several bins but concentrated in a few, the estimate for x has some reliability but is influenced by the dispersion. The reliability is moderate.
07

Compare with Databook Value

The datasbook value of 53.6 is within the range of values obtained by the students. So, the data are not in conflict with the databook value.

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Key Concepts

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

Data Organization
The first step in analyzing any dataset is to organize it neatly. Here, the dataset consists of 14 values measured by students.
We start by listing all the values: 53.8, 53.1, 56.9, 54.7, 58.2, 54.1, 56.4, 54.8, 57.3, 51.0, 55.1, 55.0, 54.2, 56.6.
This helps in understanding the range and spread of the data, making it easier to process in subsequent steps.

Effective data organization is crucial for clear and concise analysis. This initial organization also ensures that no data points are missed or duplicated.
Histogram Creation
Creating a histogram is a way to visually represent the frequency of data points within specified ranges, known as bins.
We start by deciding the range for each bin. In this case, we can use bins like:
  • 50-51
  • 51-52
  • 52-53
  • 53-54
  • 54-55
  • 55-56
  • 56-57
  • 57-58
  • 58-59
Each bin spans one unit, making it easier to count how many values fall within each range.
This step is crucial for visualizing the data and understanding its distribution.
Bin Counting
Once bins are decided, the next step is to count the number of values in each bin. This shows how frequently data points appear in each interval.
Here are the counts for each bin:
  • 50-51: 1
  • 51-52: 1
  • 52-53: 0
  • 53-54: 2
  • 54-55: 5
  • 55-56: 2
  • 56-57: 3
  • 57-58: 1
  • 58-59: 1
Counting each bin helps to create the histogram and understand the distribution pattern.
This step reveals concentrations and gaps in the data, providing insight into common and rare values.
Best Estimate
The histogram helps to identify the range within which the data is most concentrated.
For the given data, the bin with the highest frequency is 54-55, containing 5 values.
Therefore, the best estimate for the physical quantity x is around the midpoint of this bin, which is 54.5.
Choosing this value as the best estimate leverages the most common occurrence of data points within the dataset.
Data Reliability
Estimating the reliability of the chosen best value requires analyzing the spread of data.
In this case, although the data is spread across several bins, it is mainly concentrated in a few.
This moderate concentration implies that while our estimate of x being around 54.5 is reasonable, the dispersion influences its reliability.
Therefore, we conclude that the reliability of our estimate is moderate.
Value Comparison
Finally, comparing the students' data with the databook value is essential to check for consistency.
The databook gives the value of x as 53.6 with negligible error. Our students' histogram shows that 53.6 falls within the range of collected data.
This means that the students' data are not in conflict with the databook value.
The closeness of the databook value to the students' data further validates the reliability of their measurements.

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Most popular questions from this chapter

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