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3.36 Construct a histogram corresponding to each of the five frequency distributions, \(\mathrm{I}-\mathrm{V},\) given in the following table, and state whether each histogram is symmetric bimodal, positively skewed, or negatively skewed: \begin{tabular}{cccccc} & \multicolumn{5}{c} { Frequency } \\ \cline { 2 - 6 } Class Interval & I & II & III & IV & V \\ 0 to \(<10\) & 5 & 40 & 30 & 15 & 6 \\ 10 to \(<20\) & 10 & 25 & 10 & 25 & 5 \\ 20 to \(<30\) & 20 & 10 & 8 & 8 & 6 \\ 30 to \(<40\) & 30 & 8 & 7 & 7 & 9 \\ 40 to \(<50\) & 20 & 7 & 7 & 20 & 9 \\ 50 to \(<60\) & 10 & 5 & 8 & 25 & 23 \\ 60 to \(<70\) & 5 & 5 & 30 & 10 & 42 \\ \hline \end{tabular}

Short Answer

Expert verified
The student would need to provide five histograms, each based upon the frequency distributions provided, and an explanation based on their observations of the graphs illuminating whether each histogram is symmetric, bimodal, positively skewed, or negatively skewed.

Step by step solution

01

Construct Histogram for Frequency Distribution I

For this task, the student would plot the given class intervals on the X-axis and the corresponding frequencies from column I on the Y-axis. Each class interval would be represented by a bar, the height of which corresponds to the given frequency. The shape of the histogram should be examined - is it symmetric, bimodal or skewed?
02

Construct Histogram for Frequency Distribution II

Repeat Step 1, but this time plot the frequencies from column II. Examine the resulting histogram's shape - is it symmetric, bimodal or skewed?
03

Construct Histogram for Frequency Distribution III

Repeat Step 1, but this time plot the frequencies from column III. Once again, the student should examine the shape of the resulting histogram.
04

Construct Histogram for Frequency Distribution IV

Repeat Step 1, but now plot the frequencies from column IV. The student should pay attention to the shape of the resulting histogram.
05

Construct Histogram for Frequency Distribution V

Finally, the student should repeat the process from Step 1 one last time, but with the frequencies from column V. The student again should examine the shape of the resulting histogram.
06

Determine Histogram Shapes

Using the constructed histograms, the student should determine whether each histogram is symmetric (appears uniformly shaped around its middle), bimodal (has two noticeable peaks), positively skewed (most data gathered towards lower values with few outliers at high values), or negatively skewed (most data gathered towards high values with few outliers at low values).

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Key Concepts

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

Frequency Distribution
A frequency distribution is a way to summarize data by showing the number of observations within each interval. In the context of histograms, these intervals are the bins represented on the X-axis. Each bin corresponds to a range of values called a class interval. The frequency, represented on the Y-axis, reflects how many data points fall within each class interval.

Creating a frequency distribution involves a few simple steps:
  • Determine the range of the data by subtracting the smallest value from the largest.
  • Decide on the number of bins, which will influence how detailed your distribution looks.
  • Calculate the bin width by dividing the data range by the number of bins.
  • Count the number of data points falling within each bin.
Once this setup is complete, you can translate this information into a histogram which provides a visual impression of the data's spread and essential trends.
Shape Analysis
When analyzing shapes in histograms, the goal is to understand the distribution of data visually. The shape can reveal important features of your data and may indicate underlying population parameters. There are a few common types of shapes to look for when analyzing histograms:

  • Symmetric: If the left and right side of the histogram are mirror images, then it indicates that the data is uniformly distributed around the center.
  • Skewed: If one tail is longer than the other, we say that the histogram is skewed either positively or negatively.
  • Bimodal: This occurs when there are two distinct peaks in the histogram, suggesting two different modes or groups within the dataset.
  • Uniform: Indicating that each interval has roughly the same frequency.
Shape analysis provides insights into data trends and potential anomalies, helping in further statistical analyses.
Symmetric Distribution
A symmetric distribution is a type of data distribution where the left half of the graph is a mirror image of the right half. This balance means the central tendency, typically mean and median, tend to coincide. When looking at a histogram, a symmetric distribution appears bell-shaped, and it often signifies a normal distribution pattern.

Understanding a symmetric distribution is key because:
  • It suggests there are as many values lower than the mean as there are higher.
  • This balance in data can simplify further statistical inference, like calculating probabilities.
  • Many statistical tests assume the data follows a symmetric or normal distribution.
In practical terms, a symmetric histogram simplifies the analysis, as it often points to a normal, Gaussian process underlying the data.
Skewness Analysis
Skewness refers to the asymmetry in the distribution of data, and it can be either positive or negative.
  • Positive skew (right-skewness): The tail on the right side is longer or fatter than the left side. Most values accumulate towards the lower end with a few high-value outliers.
  • Negative skew (left-skewness): The tail on the left side is longer or fatter than the right side. Most data values gather at higher numbers, with a few low-value outliers.
Understanding skewness is crucial because:
  • It can affect mean and median, positioning the mean towards the longer tail.
  • It helps identify potential outliers and anomalies within data.
  • Knowing the skewness helps in choosing the right statistical tests and interpretations.
Being aware of skewness is vital for accurate data analysis, as it can heavily influence the results of analytical models.

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Most popular questions from this chapter

The National Confectioners Association asked 1006 adults the following question: "Do you set aside a personal stash of Halloween candy?" Fifty-five percent of those surveyed responded no, \(41 \%\) responded yes, and \(4 \%\) either did not answer the question or said they did not know (USA Today. October 22, 2009). Use the given information to construct a pie chart.

The report "Wireless Substitution: Early Release of Estimates from the National Health Interview Survey" (Center for Disease Control, 2009 ) gave the following estimates of the percentage of homes in the United States that had only wireless phone service at 6-month intervals from June 2005 to December 2008 . \begin{tabular}{lr} & Percent with Only \\ Date & Wireless Phone Service \\ \hline June 2005 & 7.3 \\ December 2005 & 8.4 \\ June 2006 & 10.5 \\ December 2006 & 12.8 \\ June 2007 & 13.6 \\ December 2007 & 15.8 \\ June 2008 & 17.5 \\ December 2008 & 20.2 \\ \hline \end{tabular} Construct a time-series plot for these data and describe the trend in the percent of homes with only wireless phone service over time. Has the percent increased at a fairly steady rate?

The accompanying frequency distribution summarizes data on the number of times smokers who had successfully quit smoking attempted to quit before their final successful attempt ("Demographic Variables, Smoking Variables, and Outcome Across Five Studies," Health Psychology [2007]: 278-287). \begin{tabular}{cc} Number of Attempts & Frequency \\ \hline 0 & 778 \\ 1 & 306 \\ 2 & 274 \\ \(3-4\) & 221 \\ 5 or more & 238 \\ \hline \end{tabular} Assume that no one had made more than 10 unsuccessful attempts, so that the last entry in the frequency distribution can be regarded as \(5-10\) attempts. Summarize this data set using a histogram. Be careful- the class intervals are not all the same width, so you will need to use a density scale for the histogram. Also remember that for a discrete variable, the bar for 1 will extend from 0.5 to 1.5. Think about what this will mean for the bars for the 3-4 group and the \(5-10\) group.

The article "Frost Belt Feels Labor Drain" (USA Today. May 1, 2008 ) points out that even though total population is increasing, the pool of young workers is shrinking in many states. This observation was prompted by the data in the accompanying table. Entries in the table are the percent change in the population of 25 - to 44 -year-olds over the period from 2000 to 2007 . A negative percent change corresponds to a state that had fewer 25 - to 44 -year-olds in 2007 than in 2000 (a decrease in the pool of young workers). a. The smallest value in the data set is -11.9 and the largest value is \(22.0 .\) One possible choice of stems for a stem-and-leaf display would be to use the tens digit, resulting in stems of \(-1,-0,0,1,\) and 2 . Notice that because there are both negative and positive values in the data set, we would want to use two 0 stems-one where we can enter leaves for the negative percent changes that are between 0 and -9.9 , and one where we could enter leaves for the positive percent changes that are between 0 and 9.9 . Construct a stem-and-leaf plot using these five stems. (Hint: Think of each data value as having two digits before the decimal place, so 4.1 would be regarded as 04.1.) b. Using two-digit stems would result in more than 30 stems, which is more than we would usually want for a stem-and-leaf display. Describe a strategy for using repeated stems that would result in a stemand-leaf display with about 10 stems. c. The article described "the frost belt" as the cold part of the country-the Northeast and Midwestnoting that states in the frost belt generally showed a decline in the number of people in the \(25-\) to 44 -year-old age group. How would you describe the group of states that saw a marked increase in the number of 25 - to 44 -year-olds?

Student loans can add up, especially for those attending professional schools to study in such areas as medicine, law, or dentistry. Researchers at the University of Washington studied medical students and gave the following information on the educational debt of medical students on completion of their residencies (Annals of Internal Medicine [March 2002]: \(384-398\) ): \begin{tabular}{cc} Educational Debt (dollars) & Relative Frequency \\ \hline 0 to \(<5000\) & .427 \\ 5000 to \(<20,000\) & .046 \\ 20,000 to \(<50,000\) & .109 \\ 50,000 to \(<100,000\) & .232 \\ 100,000 or more & .186 \\ \hline \end{tabular} a. What are two reasons that you could not use the given information to construct a histogram with the educational debt intervals on the horizontal axis and relative frequency on the \(y\) -axis? b. Suppose that no student had an educational debt of \(\$ 150,000\) or more upon completion of his or her residency, so that the last class in the relative frequency distribution would be 100,000 to \(<150,000\). Summarize this distribution graphically by constructing a histogram of the educational debt data. (Don't forget to use the density scale for the heights of the bars in the histogram, because the interval widths aren't all the same.) c. Based on the histogram of Part (b), write a few sentences describing the educational debt of medical students completing their residencies.

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