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. Stat. 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 Data for Stem-and-Leaf Display

Arrange the data set into two-digit numbers where the first digit represents the 'stem' and the second digit represents the 'leaf'. For instance, 4.6 becomes '4 | 6'.

02

Construct Stem-and-Leaf Display

Group the stems from smallest to largest and write down the applicable leaves for each stem. Every stem corresponds to a tens place, while each leaf corresponds to the units place of the flow rate.

03

Find a Typical Flow Rate (Median)

Sort the data set in ascending order and find the median value since it represents a typical flow rate. The median is the middle value in the ordered data set.

04

Assess Data Concentration

Examine the stem-and-leaf display to determine if the data is concentrated around a central stem or if it is distributed across many stems.

05

Analyze Distribution Symmetry

Check if the leaves on either side of the median are distributed similarly, indicating symmetry. If more values are on one side or the extension is uneven, the distribution is skewed.

06

Identify Outliers

Look for data points that are significantly distant from the rest of the data in the stem-and-leaf display. These points might qualify as outliers if they lie far from the majority of the data. For a more concrete identification, calculate the standard deviations or use box plot outliers detection methods.

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Key Concepts

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

Understanding Stem-and-Leaf Displays

Stem-and-leaf displays are handy tools in statistical data analysis for visualizing numerical data quickly and efficiently. Imagine you have a list of numbers, and you want to get a sense of its shape or distribution. Here is where a stem-and-leaf display steps in. It organizes data by splitting each number into two parts: a "stem" consisting of all but the final digit, and a "leaf", the final digit itself. For example, the number 4.6 is displayed as '4 | 6' where '4' is the stem and '6' is the leaf. This method allows you to see both the general pattern (the stem indicates the primary grouping) and the detail (the leaves indicate finer increments).
Such a display is particularly useful because it keeps the original data values intact, unlike a histogram which groups them into bins losing the specific values. You also get an immediate visual sense of the density and spread of your data, plus it's easy to spot the range, median, mode, and possible outliers.

Exploring Flow Rate Data

Flow rate data in this context refers to the measurement of water flow, expressed in liters per minute (L/min), from showers in various homes. Analyzing such data is crucial for understanding patterns in water usage across households. It provides insights into how household water consumption varies, which can be significant for researchers or policy makers interested in water conservation.
When you receive flow rate data like this, you initially organize it to make it interpretable. You can use statistical tools like a stem-and-leaf display to summarize the data. Such data organization helps in uncovering trends — whether certain flow rates are more common, identifying typical values, understanding spread and variability, and detecting unusual values that could signify errant measurements or unique conditions in the surveyed homes. This approach ensures a systematic assessment and facilitates further statistical analysis like identifying the mean and median flow rates or discovering potential anomalies.

Examining Distribution Symmetry

Distribution symmetry in data analysis helps us understand how the data values arrange themselves around the central value. When we say a distribution is symmetric, it implies the left side is a mirror image of the right side with respect to the center. But why does this matter? Symmetry tells us about the balance in data: a symmetric distribution suggests similar frequency of values below and above the center.
In practical terms, to check for symmetry in a stem-and-leaf display, consider the arrangement of leaves on either side of the median (middle value). If you notice that the leaves extend similarly on both sides, the distribution can be considered symmetric. However, when one side is stretched out farther or packed more than the other, it indicates skewness. For instance, if more data lies to the right of the median, the data is negatively skewed, and if to the left, it is positively skewed. Identifying this helps tailor the further steps in analysis, like adjustments for normalization in model predictions.

Detecting Outliers in Data

In statistical data analysis, outliers are those exceptional values that stand out from the rest of your dataset. They are important as they can significantly affect the results and interpretations of your data analysis if not properly accounted for. Outliers might result from errors in measurement or may indicate a novel phenomenon.
Detecting outliers in a stem-and-leaf display simply requires you to spot data points that deviate substantially from the bulk of other values. Often, they can appear as isolated leaves far from the condensed core of the display. To solidify the detection, statistical tools like box plots, which use interquartile ranges, or more advanced techniques such as Z-scores, can quantify how far a point is from the main distribution. Understanding outliers is critical before delving further into statistical modeling or inference, as addressing them appropriately ensures cleaner data and more valid conclusions.