91Ó°ÊÓ

The accompanying data set consists of observations on shower-flow rate (L/min) for a sample of \(n=129\) houses in Perth, Australia ("An Application of Bayes Methodology to the Analysis of Diary Records in a Water Use Study," J. Amer. Statist. Assoc., 1987: 705-711): \(\begin{array}{rrrrrrrrrr}4.6 & 12.3 & 7.1 & 7.0 & 4.0 & 9.2 & 6.7 & 6.9 & 11.5 & 5.1 \\ 11.2 & 10.5 & 14.3 & 8.0 & 8.8 & 6.4 & 5.1 & 5.6 & 9.6 & 7.5 \\\ 7.5 & 6.2 & 5.8 & 2.3 & 3.4 & 10.4 & 9.8 & 6.6 & 3.7 & 6.4 \\ 8.3 & 6.5 & 7.6 & 9.3 & 9.2 & 7.3 & 5.0 & 6.3 & 13.8 & 6.2 \\ 5.4 & 4.8 & 7.5 & 6.0 & 6.9 & 10.8 & 7.5 & 6.6 & 5.0 & 3.3 \\ 7.6 & 3.9 & 11.9 & 2.2 & 15.0 & 7.2 & 6.1 & 15.3 & 18.9 & 7.2 \\ 5.4 & 5.5 & 4.3 & 9.0 & 12.7 & 11.3 & 7.4 & 5.0 & 3.5 & 8.2 \\ 8.4 & 7.3 & 10.3 & 11.9 & 6.0 & 5.6 & 9.5 & 9.3 & 10.4 & 9.7 \\ 5.1 & 6.7 & 10.2 & 6.2 & 8.4 & 7.0 & 4.8 & 5.6 & 10.5 & 14.6 \\ 10.8 & 15.5 & 7.5 & 6.4 & 3.4 & 5.5 & 6.6 & 5.9 & 15.0 & 9.6 \\ 7.8 & 7.0 & 6.9 & 4.1 & 3.6 & 11.9 & 3.7 & 5.7 & 6.8 & 11.3 \\ 9.3 & 9.6 & 10.4 & 9.3 & 6.9 & 9.8 & 9.1 & 10.6 & 4.5 & 6.2 \\ 8.3 & 3.2 & 4.9 & 5.0 & 6.0 & 8.2 & 6.3 & 3.8 & 6.0 & \end{array}\) a. Construct a stem-and-leaf display of the data. b. What is a typical, or representative, flow rate? c. Does the display appear to be highly concentrated or spread out? d. Does the distribution of values appear to be reasonably symmetric? If not, how would you describe the departure from symmetry? e. Would you describe any observation as being far from the rest of the data (an outlier)?

Step by step solution

01

- Organize the Data

First, review the data set and organize the shower flow rates. We have a total of 129 observations, ranging from the lowest to the highest values.

02

- Construct the Stem-and-Leaf Display

A stem-and-leaf display represents the data by splitting each value into a 'stem' and a 'leaf'. Here, the 'stem' consists of the integer part of the data, and the 'leaf' consists of the decimal part. For instance, the number 4.6 will have a stem of 4 and a leaf of 6.

03

- Create Stem Categories

Choose stems ranging from 2 to 18, as the data values are between 2.2 and 18.9. Each stem represents a different integer starting value: for example, 2 represents 2.0 to 2.9.

04

- Fill in the Leaves

Under each stem category, write the leaf digits for all numbers that correspond to that particular stem value in increasing order. For example, under the stem '4', list the leaf values from the numbers: 4.0, 4.1, 4.3, 4.5, etc.

05

- Analyze the Typical Flow Rate

To find a typical flow rate, look for the stems with the most dense concentration of leaves. The data appears to be most heavily concentrated around the stem values from 6 to 7, suggesting the typical flow rate is between 6.0 and 7.9 L/min.

06

- Assess Concentration

Check the distribution of leaf values across stems. If most values cluster within a few stems, the data is concentrated. The organized stem-and-leaf plot shows several values around the 6 and 7 range, indicating the data is moderately concentrated.

07

- Check Symmetry

For symmetry, observe both sides of the stem-and-leaf plot. The plot is not symmetric: there are more higher values (right-skewed) than lower values, showing a positive skewness.

08

- Identify Outliers

An outlier is a point far from the rest of the data. The values around 15 and above could be considered outliers because they stand out from the remainder of the observations clustered between stems 2 and 14.

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Key Concepts

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

Stem-and-Leaf Plot

A stem-and-leaf plot is a simple way to display data, especially useful for small datasets, like the one for shower flow rates in Perth. This type of plot allows each data point's value to be clearly seen while providing a view of the overall distribution. Each number is divided into a "stem" (the leading digit(s) of the number) and a "leaf" (the trailing digit). For instance, in the shower flow rate data, a value of 6.9 is split into a stem of 6 and a leaf of 9.

To create the plot, you begin by determining the stems, which usually range from the smallest to the largest leading digits in the data. For this particular dataset, stems range from 2 to 18, which conveniently captures all the data points as the shower flow rates go from 2.2 to 18.9 L/min. After setting up the stems, each flow rate is assigned to the appropriate stem and arranged in order, effectively showing the spread and concentration of values.

• The stem-and-leaf plot condenses the data.
• Displays the frequency of data distribution.
• Helps easily identify the shape and spread of the dataset.

Data Symmetry

Data symmetry in statistics refers to how evenly the values are distributed around the center of the dataset. In a perfectly symmetric distribution, the left-hand side mirrors the right-hand side of a graph. For our shower flow rate example, symmetry is checked by visually inspecting the stem-and-leaf plot.

Upon observing, this dataset does not seem to represent a symmetrical pattern. Instead, it exhibits a positive skew, meaning there are more values on the lower end, stretching less towards the higher end. This skewness is evident because the stems, particularly those around 6 and 7 which hold the most data, show more leaves on the higher side instead of being balanced.

• Symmetry implies a balanced dataset.
• Skewness indicates more values are stacked on one side.
• Positive skew (right-skewed) suggests a tail on the right.

Outliers

Outliers are data points that differ significantly from other observations. They can indicate variability, errors, or new insights. In our shower flow rate dataset, identifying outliers involves examining extreme values that fall outside the usual pattern observed in the stem-and-leaf plot.

Typically, an outlier resides far from the main concentration of data points. In this dataset, flow rates like 15.0 L/min and above are noticeable, as they fall beyond the range where most other data values reside (clusters between stems 6 to 9). Such outliers can skew the analysis and should be considered when making interpretations.

• Outliers can affect mean and standard deviation.
• Might indicate errors or unusual cases.
• Impact the symmetry and concentration of data.

Shower Flow Rate Analysis

The analysis of shower flow rates can inform water usage policies and resource management, particularly in urban areas. Through descriptive statistics and visual tools like the stem-and-leaf plot, understanding distribution and outlier observations is enhanced.

For example, this data set can offer insights into typical water use patterns in Perth homes, with most shower flows ranging from 6.0 to 7.9 L/min, as per the plot's concentrated area. Recognizing typical usage helps in forecasting demands and drafting water conservation measures. Noting any deviations like abnormally high usage through outliers can pinpoint inefficiencies or singular events worth investigating.

• Establishes typical usage benchmarks.
• Informs resource allocation strategies.
• Highlights the need for further investigation into unusual patterns.