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Chapter 1: Problem 19

Aqua running has been suggested as a method of exercise for injured athletes and others who want a low-impact aerobics program. A study reported in the Journal of Sports Medicine reported the heart rates of 20 healthy volunteers at a cadence of 96 steps per minute. \({ }^{8}\) The data are listed here: \(\begin{array}{rrrrrrrrrr}87 & 109 & 79 & 80 & 96 & 95 & 90 & 92 & 96 & 98 \\\ 101 & 91 & 78 & 112 & 94 & 98 & 94 & 107 & 81 & 96\end{array}\) a. Construct a stem and leaf plot to describe the data. b. Discuss the characteristics of the data distribution.

Short Answer

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a) Skewed right, centered around 95, large spread, no outliers b) Skewed left, centered around 90, small spread, at least one outlier c) Skewed right, centered around 80, large spread, at least one outlier Answer: a) Skewed right, centered around 95, large spread, no outliers

Step by step solution

01

Construct a Stem and Leaf Plot

First, we need to sort the data in ascending order: 78, 79, 80, 81, 87, 90, 91, 92, 94, 94, 95, 96, 96, 96, 98, 98, 101, 107, 109, 112 Now we can create a stem and leaf plot where the stems are the tens digits, and the leaves are the units digits. Here's the stem and leaf plot: Stem | Leaves ------------------ 7 | 8 9 8 | 0 1 7 9 | 0 1 2 4 4 5 6 6 6 8 8 10 | 1 7 9 11 | 2
02

Describe the Characteristics of the Data Distribution

Now that we have the stem and leaf plot, we can analyze the characteristics of the data distribution: 1) Shape: The distribution appears to be slightly skewed to the right, as the right tail seems longer than the left tail. Most of the heart rates are in the 90s. 2) Center: The data seems to have a center around 95, as that is approximately the value where half of the data lies above and half lies below. 3) Spread: The data ranges from a minimum of 78 to a maximum of 112. There is a relatively large spread in the data, which could be an indication of a high variability. 4) Outliers: There doesn't seem to be any apparent outliers, as the values are all relatively close to one another. In conclusion, the stem and leaf plot of the heart rates of healthy volunteers during Aqua running shows a slightly skewed right distribution, centered around 95, with a relatively large spread and no apparent outliers.

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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 organize data and see its distribution quickly. It's especially useful for small data sets. Let's look at how this works:

- **Stems represent the tens place**: This is where we group data together by their tens digit. For example, both 78 and 79 would be grouped under the stem '7'.
- **Leaves represent the units place**: Each leaf shows the individual data points' units digit. For instance, if we have the data 78 and 79 under the stem '7', the leaves would be '8' and '9'.

By arranging our data from lowest to highest, we create a visual like this:
  • 7 | 8 9
  • 8 | 0 1 7
  • 9 | 0 1 2 4 4 5 6 6 6 8 8
  • 10 | 1 7 9
  • 11 | 2
This method of visualization helps us easily see patterns, like gaps or clusters, without complicated calculations.
Data Distribution
Data distribution refers to how data values are spread out or clustered over a range. By having our stem and leaf plot above, we can identify how these values are distributed.

- The data mainly clusters around the values in the 90s, indicating that many participants' heart rates during Aqua running fell into this range.
- Notice how some values are higher, reaching the low 100s, or lower in the 70s, showing variation among the participants.

Distributions can be symmetrical, where data is evenly spread around a central value, or skewed, where there is a tail on one side, affecting the overall shape. Recognizing the distribution helps understand the underlying trends and variations.
Skewness
Skewness is a measure of the asymmetry in the distribution of data. It tells us about the direction and extent of the tail. In our case, the data is right-skewed:

- **Right skewed** means the majority of data points are concentrated to the left, with the tail extending to the right. This is illustrated by more data points around the 90s, tailing off to the higher numbers like 112.
- Such skewness could be due to several factors, including varied fitness levels of participants.

Understanding skewness helps in explaining whether extreme values (like very high heart rates) influence the average and are typical or unusual in the given context.
Center and Spread
The center of the data is a value around which data values tend to cluster. In our exercise, the center is approximately at 95, indicating that, on average, participants' heart rates hovered around this number during Aqua running.

The spread tells us how much the data varies:

- **Range**: This is simply the difference between the highest and lowest values (112 - 78 = 34).
- A large spread like this suggests high variability among heart rates, which might result from individual differences in fitness or measurement conditions.
- By analyzing both center and spread, we gain insights into not just what typical values might be, but also about the diversity or consistency of the set as a whole.

Together, these metrics provide a comprehensive picture of our data set, aiding in a holistic understanding of the patterns present.

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Most popular questions from this chapter

Determine whether the data collected represents a population or a sample. Twenty animals are put on a new diet and their weight gain over 3 months is recorded.

A discrete variable can take on only the values \(0,1,\) or \(2 .\) Use the set of 20 measurements on this variable to answer the questions. $$ \begin{array}{lllll} 1 & 2 & 1 & 0 & 2 \\ 2 & 1 & 1 & 0 & 0 \\ 2 & 2 & 1 & 1 & 0 \\ 0 & 1 & 2 & 1 & 1 \end{array} $$ Compare the dotplot and the stem and leaf plot. Do they convey roughly the same information?

Construct a relative frequency histogram for these 50 measurements using classes starting at 1.6 with a class width of .5. Then answer the questions. $$\begin{array}{llllllllll}3.1 & 4.9 & 2.8 & 3.6 & 2.5 & 4.5 & 3.5 & 3.7 & 4.1 & 4.9 \\ 2.9 & 2.1 & 3.5 & 4.0 & 3.7 & 2.7 & 4.0 & 4.4 & 3.7 & 4.2 \\ 3.8 & 6.2 & 2.5 & 2.9 & 2.8 & 5.1 & 1.8 & 5.6 & 2.2 & 3.4 \\ 2.5 & 3.6 & 5.1 & 4.8 & 1.6 & 3.6 & 6.1 & 4.7 & 3.9 & 3.9 \\ 4.3 & 5.7 & 3.7 & 4.6 & 4.0 & 5.6 & 4.9 & 4.2 & 3.1 & 3.9\end{array}$$ What fraction of the measurements are less than \(5.1 ?\)

Create a dotplot for the number of cheeseburgers eaten in a given week by 10 college students. \(\begin{array}{llll}4 & 5 & 4 & 2\end{array}\) \(\begin{array}{lllll}3 & 3 & 4 & 2 & 7\end{array}\) a. How would you describe the shape of the distribution? b. What proportion of the students ate more than 4 cheeseburgers that week?

The ages (in months) at which 50 children were first enrolled in a preschool are listed as follows. $$ \begin{array}{llllllllll} 38 & 40 & 30 & 35 & 39 & 40 & 48 & 36 & 31 & 36 \\ 47 & 35 & 34 & 43 & 41 & 36 & 41 & 43 & 48 & 40 \\ 32 & 34 & 41 & 30 & 46 & 35 & 40 & 30 & 46 & 37 \\ 55 & 39 & 33 & 32 & 32 & 45 & 42 & 41 & 36 & 50 \\ 42 & 50 & 37 & 39 & 33 & 45 & 38 & 46 & 36 & 31 \end{array} $$ a. Construct a relative frequency histogram for these data. Start the lower boundary of the first class at 30 and use a class width of 5 months. b. What proportion of the children were 35 months or older, but less than 45 months of age when first enrolled in preschool? c. If one child were selected at random from this group of children, what is the probability that the child was less than 50 months old when first enrolled in preschool?
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