Question 1: The Following Information Regarding The Ten Rich ✓ Solved

Question 1: The following information regarding the ten rich

Question 1: The following information regarding the ten richest Americans was reported in a recent issue of Forbes.

Question 1a: How many elements are in the above data set?

Question 1b: How many variables are in this data set?

Question 1c: How many observations are in this data set?

Question 1d: Which variables are categorical and which are quantitative?

Question 2: A sample of the ages of 10 employees of a company is shown below. Using a method of your choosing, construct a dot plot for the above data.

Question 3: The following tabular data show majors, school type, cost, 30-year ROI, and annual ROI for multiple colleges. Summarize the variables, identify their types, and describe appropriate graphical displays and summary statistics to compare majors and school types.

Paper For Above Instructions

Executive summary

This paper answers the three assignment questions using standard statistical definitions and best practices. For Question 1 I interpret the brochure-style Forbes report and explain how to count elements, variables, and observations and how to classify variables as categorical or quantitative. For Question 2 I explain how to create a dot plot, provide an illustrative dot plot for a plausible 10-age sample, and explain interpretation. For Question 3 I summarize the college ROI tabular data (majors, school type, cost, 30-year ROI, annual ROI), identify variable types, and recommend graphical displays and summary statistics to compare majors and school types. Recommendations reference established statistical and visualization guidance (Tukey, 1977; Cleveland, 1993; Moore et al., 2017).

Question 1 — Counting and classifying elements, variables, and observations

Definitions: In descriptive statistics, an element (or unit) refers to a single entity in the dataset (for example, one person), a variable is a measurable or categorical attribute recorded for each element (for example, net worth), and an observation is the recorded values of all variables for one element (Moore et al., 2017).

1a. How many elements are in the above data set?

The dataset is described as "the ten richest Americans." Therefore the number of elements is 10 (one element per person listed) (Forbes, n.d.).

1b. How many variables are in this data set?

The precise number of variables depends on how Forbes presented the list. Typical Forbes entries include: rank, name, net worth (numeric), primary source or industry (categorical), age (numeric), and residence or country (categorical). A minimally reported table might have 4–6 variables; if the report shows rank, name, net worth, source, and age, that is five variables. Without the original table we state the general rule: count each column/attribute as a variable (Moore et al., 2017).

1c. How many observations are in this data set?

Each element produces one observation. For ten individuals, there are 10 observations (one per person) (Moore et al., 2017).

1d. Which variables are categorical and which are quantitative?

Common classifications for such a list are:

  • Categorical variables: name (identifier), primary source/industry (e.g., technology, finance), residence/state (nominal); rank is ordinal categorical.
  • Quantitative variables: net worth (continuous or large discrete dollars), age (discrete numeric).

The rule is: variables representing measurements or counts are quantitative; variables representing labels or groups are categorical (McClave et al., 2018).

Question 2 — Constructing a dot plot (method and illustrative example)

Method: A dot plot shows each observation as a dot plotted on a single axis (typically the x-axis for numeric values). To create one: list the unique values along the numeric axis (ages), then stack a dot for each observation above the corresponding value. Dot plots are particularly useful for small samples (n≈10–30) because they show individual values and clustering (Tukey, 1977; Cleveland, 1993).

Note on the provided assignment: the prompt states a 10-age sample is "shown below" but does not include the numerical ages. Because the data were not provided, I demonstrate the method with a plausible illustrative sample of 10 ages and construct the dot plot for that sample. If you provide the original ages, apply the same procedure below to your data.

Illustrative sample (n = 10): 23, 25, 29, 34, 34, 40, 45, 45, 52, 57.

Dot plot (illustrative):

Age:   20  22  24  26  28  30  32  34  36  38  40  42  44  46  48  50  52  54  56  58


Dots:          ·   ·       ·   ·       ··      ··               ·       ·


Key: each "·" = one employee

Interpretation: From this illustrative dot plot we can see a concentration in the 30–46 range, two employees at age 34 and two at 45, and a right tail up to age 57. For actual analyses, compute mean, median, range, and standard deviation to summarize central tendency and dispersion (Moore et al., 2017).

Question 3 — Summarizing the college ROI tabular data and recommended analyses

Dataset description (from prompt): the table includes rows of colleges with fields: major (Business or Engineering), school type (Private or Public), cost (numeric, dollars), 30-year ROI (numeric, dollars), and annual ROI (percentage). These are typical variables when comparing college returns (Payscale, 2013).

Variable types and structure:

  • Major: categorical (nominal: Business, Engineering)
  • School Type: categorical (nominal: Private, Public)
  • Cost: quantitative (continuous; measured in dollars)
  • 30-Year ROI: quantitative (continuous; dollars)
  • Annual ROI: quantitative (percentage; continuous)

Recommended graphical displays:

  • Boxplots of 30-year ROI and Annual ROI split by Major and by School Type to compare distributions and identify outliers (Tukey, 1977).
  • Side-by-side bar charts (or mean ± standard error bars) of mean annual ROI by Major and School Type to show group-level comparisons (Cleveland, 1993).
  • Scatter plot of Cost versus 30-year ROI with points colored by Major and shaped by School Type to visualize relationships and detect cost/return tradeoffs (Tufte, 2001).
  • Histogram of annual ROI values to inspect skewness and distributional shape.

Recommended summary statistics and tests:

  • For each group (e.g., Business-Private, Business-Public, Engineering-Private, Engineering-Public): report n, mean, median, standard deviation, interquartile range (IQR), min and max (Moore et al., 2017).
  • Use t-tests or nonparametric equivalents (Mann–Whitney U) to compare two groups (e.g., Private vs Public within a major). For comparing more than two groups (e.g., all four group combinations), use ANOVA with post-hoc comparisons; check homogeneity of variance and normality assumptions (NIST/SEMATECH, n.d.; UCLA IDRE, n.d.).
  • Compute effect sizes (Cohen’s d or eta-squared) to quantify practical significance beyond p-values.

Example interpretation guidance: If Engineering-Private shows a higher median 30-year ROI than Business-Public (and the difference is statistically significant with a medium-to-large effect size), one might conclude that, on average, private engineering programs in this sample produce higher long-term monetary returns. However, check cost differences: a much higher upfront cost may reduce net ROI. Use the scatter plot to visualize this trade-off.

Concluding recommendations

1. For Question 1: answer element = 10 and observations = 10; variables must be counted from the original Forbes table (commonly 4–6). Classify name and source as categorical; net worth and age as quantitative.

2. For Question 2: produce a dot plot for the exact ages by stacking dots at each value. The illustrative dot plot shown demonstrates the procedure for a 10-value sample. Use dot plots for small samples and report simple summary stats.

3. For Question 3: treat Major and School Type as categorical and Cost, 30-year ROI, Annual ROI as quantitative. Use boxplots, scatter plots, histograms, and groupwise summary statistics and inferential tests (t-test/ANOVA) to compare groups and quantify differences, while reporting effect sizes and checking assumptions.

References

  • Forbes. (n.d.). The Forbes billionaires list. Forbes. https://www.forbes.com (example source for wealthy individual listings)
  • Payscale. (2013). Best college ROI by major. Payscale.com. https://www.payscale.com
  • Moore, D. S., McCabe, G. P., & Craig, B. (2017). Introduction to the Practice of Statistics (9th ed.). W. H. Freeman.
  • McClave, J. T., Benson, P. G., & Sincich, T. (2018). Statistics for Business and Economics (13th ed.). Pearson.
  • Tukey, J. W. (1977). Exploratory Data Analysis. Addison-Wesley.
  • Cleveland, W. S. (1993). Visualizing Data. Hobart Press.
  • Tufte, E. R. (2001). The Visual Display of Quantitative Information (2nd ed.). Graphics Press.
  • NIST/SEMATECH. (n.d.). e-Handbook of Statistical Methods. National Institute of Standards and Technology. https://www.itl.nist.gov/div898/handbook
  • UCLA Institute for Digital Research and Education (IDRE). (n.d.). UCLA Statistical Consulting Group: Resources for data visualization and analysis. https://stats.idre.ucla.edu
  • OpenIntro. (n.d.). OpenIntro Statistics. https://www.openintro.org

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