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Chapter 1: Problem 3

The accompanying summary data on \(\mathrm{CeO}_{2}\) particle sizes \((\mathrm{nm}\) ) under certain experimental conditions was read from a graph in the article "NanoceriaEnergetics of Surfaces, Interfaces and Water Adsorption" (J. of the Amer. Ceramic Soc., 2011: 3992-3999): \(\begin{array}{ccccc}3.0-<3.5 & 3.5-<4.0 & 4.0-<4.5 & 4.5-<5.0 & 5.0-5.5 \\ 5 & 15 & 27 & 34 & 22\end{array}\) \(\begin{array}{ccccc}5.5-<6.0 & 6.0-<6.5 & 6.5-<7.0 & 7.0-<7.5 & 7.5-<8.0 \\\ 14 & 7 & 2 & 4 & 1\end{array}\) a. What proportion of the observations are less than 5 ? b. What proportion of the observations are at least 6 ? c. Construct a histogram with relative frequency on the vertical axis and comment on interesting features. In particular, does the distribution of particle sizes appear to be reasonably symmetric or somewhat skewed? [Note: The investigators fit a lognormal distribution to the data; this is discussed in Chapter 4.] d. Construct a histogram with density on the vertical axis and compare to the histogram in (c).

Short Answer

Expert verified
a) 0.618, b) 0.214, c & d) Distribution is right-skewed.

Step by step solution

01

Understanding the Frequency Data

We are provided with frequency data on particle sizes, grouped into intervals. The intervals are: \(3.0-<3.5\), \(3.5-<4.0\), \(4.0-<4.5\), \(4.5-<5.0\), \(5.0-<5.5\), \(5.5-<6.0\), \(6.0-<6.5\), \(6.5-<7.0\), \(7.0-<7.5\), and \(7.5-<8.0\). The respective frequencies are: 5, 15, 27, 34, 22, 14, 7, 2, 4, and 1.
02

Total Observations

Add all the frequencies to determine the total number of observations. \(5 + 15 + 27 + 34 + 22 + 14 + 7 + 2 + 4 + 1 = 131\). Thus, the total number of observations is 131.
03

Proportion Less than 5

Sum the frequencies of intervals less than 5: \(5 + 15 + 27 + 34 = 81\). Then, find the proportion: \(\frac{81}{131}\). This is approximately 0.618.
04

Proportion at Least 6

Sum the frequencies of intervals at least 6: \(14 + 7 + 2 + 4 + 1 = 28\). Then, find the proportion: \(\frac{28}{131}\). This is approximately 0.214.
05

Construct Histogram with Relative Frequency

To construct a histogram with relative frequency, divide the frequency of each interval by the total number of observations and plot these values against the intervals. The relative frequencies are: \(\frac{5}{131}, \frac{15}{131}, \frac{27}{131}, \frac{34}{131}, \frac{22}{131}, \frac{14}{131}, \frac{7}{131}, \frac{2}{131}, \frac{4}{131}, \frac{1}{131}\).
06

Symmetry or Skewness Analysis

By observing the histogram with relative frequency, note that the data is not symmetric. It is right-skewed since there are more observations at lower particle sizes and fewer at higher sizes.
07

Construct Histogram with Density

To create a histogram with density, determine the density for each interval by dividing the relative frequency by the interval width (0.5 nm for each). Plot these densities against the intervals to create the density histogram.
08

Compare Histograms

Compare the histogram with density to the one with relative frequency. While the shape remains similar, densities allow the area of each bar to sum to 1, offering a normalized view of the data. Both histograms show the right-skewed nature of the data.

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Key Concepts

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

Relative Frequency
When constructing a histogram, a crucial first step involves calculating the relative frequency of each data interval. This metric tells us how often a particular data value appears relative to the entire dataset. To find the relative frequency for each interval, simply divide the frequency of that interval by the total number of observations. For example, if an interval has a frequency of 5 and the total number of observations is 131, the relative frequency is calculated as \( \frac{5}{131} \). Moreover, this relative frequency is expressed as a fraction or a percentage.
Plotting these relative frequencies on the vertical axis of a histogram visualizes how data is distributed across intervals. It provides a clearer comparison between different intervals because it standardizes the data, allowing each interval's contribution to the overall dataset to be universally comparable.
Density Histogram
A density histogram offers an insightful way to visualize data by normalizing it so that the area of the bars sums to 1. To construct it, calculate each interval's density by taking its relative frequency and dividing it by the interval width. For example, an interval with a relative frequency of \( \frac{5}{131} \) and width of 0.5 nm would have a density of \( \frac{5/131}{0.5} \).
This methodology is particularly useful because it adjusts for varying interval widths, making each bar directly comparable across varying sizes. The density histogram provides more than just visual appeal; it aids in understanding the probability distribution of continuous data. Overlays of density curves, like the normal distribution, help in assessing the fit of our data shape to known statistical models, such as in the often-used comparison with a normal or lognormal distribution.
Data Symmetry
In statistical data analysis, recognizing whether a dataset is symmetric is essential for choosing the right statistical methods. Symmetric data will exhibit a mirror-like distribution around the center, often resembling a bell curve.
However, when we observe the histogram constructed with our data, it is clear that the data is not symmetric. This visualization is crucial, as non-symmetric datasets can hint at underlying processes governing data distribution or specific skewness. Recognizing whether data are symmetric or not helps in deciding the appropriate statistical tests to perform and in anticipating potential skewness in data distribution.
Skewness Analysis
Understanding skewness is vital for interpreting data distribution. Skewness refers to the degree of asymmetry in a distribution. If a distribution is perfectly symmetrical, it has zero skewness.
However, our analysis reveals a right-skewed (positively skewed) distribution; more data points cluster around the lower range, with a tail extending to the right. This skewness suggests that while most particle sizes are small, a few larger particles pull the tail to the right. Such information is critical for both statistical analysis and practical interpretations, as skewness can have implications for average measures, variability, and in this case, possibly some of the underlying processes in the experiment affecting particle size distribution. Understanding skewness guides on applying transformations or different statistical models to better capture and describe the dataset.

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Most popular questions from this chapter

Zinfandel is a popular red wine varietal produced almost exclusively in California. It is rather controversial among wine connoisseurs because its alcohol content varies quite substantially from one producer to another. In May 2013, the author went to the website klwines .com, randomly selected 10 zinfandels from among the 325 available, and obtained the following values of alcohol content (\%): \(\begin{array}{lllll}14.8 & 14.5 & 16.1 & 14.2 & 15.9 \\ 13.7 & 16.2 & 14.6 & 13.8 & 15.0\end{array}\) a. Calculate and interpret several measures of center. b. Calculate the sample variance using the defining formula. c. Calculate the sample variance using the shortcut formula after subtracting 13 from each observation.

Consider a sample \(x_{1}, x_{2}, \ldots, x_{n}\) and suppose that the values of \(\bar{x}, s^{2}\), and \(s\) have been calculated. a. Let \(y_{i}=x_{i}-\bar{x}\) for \(i=1, \ldots, n\). How do the values of \(s^{2}\) and \(s\) for the \(y_{i}\) 's compare to the corresponding values for the \(x_{i}\) 's? Explain. b. Let \(z_{i}=\left(x_{i}-\bar{x}\right) / s\) for \(i=1, \ldots, n\). What are the values of the sample variance and sample standard deviation for the \(z_{i} s\) ?

A certain city divides naturally into ten district neighborhoods. How might a real estate appraiser select a sample of single-family homes that could be used as a basis for developing an equation to predict appraised value from characteristics such as age, size, number of bathrooms, distance to the nearest school, and so on? Is the study enumerative or analytic?

The article "Effects of Short-Term Warming on Low and High Latitude Forest Ant Communities" (Ecoshpere, May 2011, Article 62) described an experiment in which observations on various characteristics were made using minichambers of three different types: (1) cooler (PVC frames covered with shade cloth), (2) control (PVC frames only), and (3) warmer (PVC frames covered with plastic). One of the article's authors kindly supplied the accompanying data on the difference between air and soil temperatures \(\left({ }^{\circ} \mathrm{C}\right)\). \(\begin{array}{ccc}\text { Cooler } & \text { Control } & \text { Warmer } \\\ 1.59 & 1.92 & 2.57 \\ 1.43 & 2.00 & 2.60 \\ 1.88 & 2.19 & 1.93 \\ 1.26 & 1.12 & 1.58 \\ 1.91 & 1.78 & 2.30 \\ 1.86 & 1.84 & 0.84 \\ 1.90 & 2.45 & 2.65 \\ 1.57 & 2.03 & 0.12 \\ 1.79 & 1.52 & 2.74 \\ 1.72 & 0.53 & 2.53 \\\ 2.41 & 1.90 & 2.13 \\ 2.34 & & 2.86 \\ 0.83 & & 2.31 \\ 1.34 & & 1.91 \\\ 1.76 & & \end{array}\) a. Compare measures of center for the three different samples. b. Calculate, interpret, and compare the standard deviations for the three different samples. c. Do the fourth spreads for the three samples convey the same message as do the standard deviations about relative variability? d. Construct a comparative boxplot (which was included in the cited article) and comment on any interesting features.

The accompanying frequency distribution of fracture strength (MPa) observations for ceramic bars fired in a particular kiln appeared in the article "Evaluating Tunnel Kiln Performance" (Amer. Ceramic Soc. Bull., Aug. 1997: 59-63). \(\begin{array}{lcccccc}\text { Class } & 81-<83 & 83-<85 & 85-<87 & 87-<89 & 89-<91 \\ \text { Frequency } & 6 & 7 & 17 & 30 & 43 \\ \text { Class } & 91-<93 & 93-<95 & 95-<97 & 97-<99 \\ \text { Frequency } & 28 & 22 & 13 & 3\end{array}\) a. Construct a histogram based on relative frequencies, and comment on any interesting features. b. What proportion of the strength observations are at least 85 ? Less than 95 ? c. Roughly what proportion of the observations are less than \(90 ?\)
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