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Chapter 3: Problem 20

The following data were used to construct the histograms of the number of days required to fill orders for Dawson Supply, Inc., and J.C. Clark Distributors (see Figure 3.2). \(\begin{array}{lcccccccccc}\text {Dawson Supply Days for Delivery:} & 11 & 10 & 9 & 10 & 11 & 11 & 10 & 11 & 10 & 10 \\ \text {Clark Distributors Days for Delivery:} & 8 & 10 & 13 & 7 & 10 & 11 & 10 & 7 & 15 & 12\end{array}\) Use the range and standard deviation to support the previous observation that Dawson Supply provides the more consistent and reliable delivery times.

Short Answer

Expert verified
Dawson Supply is more consistent based on its smaller range and lower standard deviation.

Step by step solution

01

Calculate the Range for Dawson Supply

First, identify the maximum and minimum values for Dawson Supply. From the data given, the maximum delivery days is 11 and the minimum is 9. The range is calculated as: \( \text{Range} = \text{Maximum} - \text{Minimum} = 11 - 9 = 2. \)
02

Calculate the Range for Clark Distributors

Similarly, determine the maximum and minimum values for Clark Distributors. The maximum delivery days is 15 and the minimum is 7. The range is therefore: \( \text{Range} = 15 - 7 = 8. \)
03

Interpret the Range

A smaller range indicates more consistency. Dawson Supply has a range of 2, which is smaller than Clark Distributors' range of 8, suggesting Dawson has more consistent delivery times.
04

Calculate the Mean for Dawson Supply

Calculate the mean to use in the standard deviation calculation. The mean is the sum of delivery days divided by the number of data points: \( \text{Mean}_{Dawson} = \frac{11 + 10 + 9 + 10 + 11 + 11 + 10 + 11 + 10 + 10}{10} = 10.3. \)
05

Calculate the Standard Deviation for Dawson Supply

Compute the standard deviation using the formula: \( \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}} \), where \( \mu \) is the mean, and \( N \) is the number of points. For Dawson Supply: \( \sigma_{Dawson} = \sqrt{\frac{(11-10.3)^2 + (10-10.3)^2 + ... + (10-10.3)^2}{10}} \approx 0.823. \)
06

Calculate the Mean for Clark Distributors

Similarly, compute Clark Distributors' mean: \( \text{Mean}_{Clark} = \frac{8 + 10 + 13 + 7 + 10 + 11 + 10 + 7 + 15 + 12}{10} = 10.3. \)
07

Calculate the Standard Deviation for Clark Distributors

Proceed with the standard deviation for Clark Distributors: \( \sigma_{Clark} = \sqrt{\frac{(8-10.3)^2 + (10-10.3)^2 + ... + (12-10.3)^2}{10}} \approx 2.58. \)
08

Compare Standard Deviations

Standard deviation measures variability around the mean. Dawson Supply's standard deviation is approximately 0.823, while Clark's is approximately 2.58. Thus, Dawson's delivery times are more consistent and reliable.

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Key Concepts

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

Range in Statistics
The range is a basic statistical concept that provides a simple measure of the spread or dispersion of a dataset. It is calculated by subtracting the smallest value in the dataset from the largest value.

For example, if we have a set of delivery days like the one from Dawson Supply with maximum days at 11 and minimum at 9, the range becomes:
  • Range = Maximum - Minimum
  • Range = 11 - 9
  • Range = 2
This tells us that the number of days required to fill orders in this set varies by up to 2 days.

When comparing ranges from two datasets, a smaller range suggests more consistency because fewer variations occur between the highest and the lowest values. In our case, Dawson Supply's lower range compared to Clark Distributors (where the range is 15 - 7 = 8) implies that Dawson's delivery times are more consistent.
Understanding Standard Deviation
The standard deviation is a crucial statistical measure that describes how spread out the values in a dataset are around the mean. Calculating it requires several steps and begins with finding the mean of the dataset.

For Dawson Supply:
  • Mean = (11 + 10 + 9 + 10 + 11 + 11 + 10 + 11 + 10 + 10) / 10 = 10.3
Next, for each data point, we find the squared difference from the mean, sum up these squared differences, and finally, divide by the number of points before taking the square root:
  • Standard Deviation = \( \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}} \)
  • \( \sigma_{Dawson} \approx 0.823 \)
This outcome indicates overall, how much variation or "noise" there is relative to the average delivery days. A smaller standard deviation like Dawson Supply鈥檚 suggests the delivery times cluster closely around the mean, highlighting reliability.

On the other hand, Clark Distributors has a higher standard deviation of approximately 2.58, indicating more variability in delivery times, thus less reliability.
Calculating the Mean
The mean, often referred to as the average, is one of the most fundamental statistics. It's used to quickly gauge the central tendency of a dataset.

To calculate it, add all the values together and divide by the total number of values. For instance, for Dawson Supply:
  • Total of delivery days: 11 + 10 + 9 + 10 + 11 + 11 + 10 + 11 + 10 + 10 = 103
  • Number of data points: 10
  • Mean = Total / Number of points = 103 / 10 = 10.3
This mean is basically the typical delivery time you can expect. Furthermore, comparing mean values helps understand whether there is a significant difference in the central tendency between two groups.

For both Dawson Supply and Clark Distributors, the mean is 10.3, indicating that on average, both deliver within similar timeframes. However, as shown by the range and standard deviation, the consistency and reliability differ between the two, marking Dawson Supply as more dependable.

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Most popular questions from this chapter

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