Math Problem Statement

A random sample of 40 college professors is selected from all professors at a university. The following list gives their ages. 65​, 48​, 41​, 47​, 30​, 52​, 40​, 42​, 41​, 23​, 53​, 66​, 52​, 49​, 67​, 62​, 59​, 46​, 56​, 28​, 34​, 61​, 47​, 41​, 30​, 39​, 41​, 36​, 21​, 41​, 61​, 30​, 36​, 43​, 22​, 33​, 41​, 34​, 33​, 55.

Construct a​ stem-and-leaf plot for the data. Complete parts a. and b.

Solution

Constructing the Stem-and-Leaf Plot

A stem-and-leaf plot organizes data by dividing each number into two parts: the stem (typically the leading digits) and the leaf (the trailing digit). For the provided data:

Steps:

  1. Separate the numbers: Use the tens place as the stem and the ones place as the leaf.

    • For example, 6565 has a stem of 66 and a leaf of 55.
  2. Group by stems: Organize the numbers into their respective stems.

  3. Order the leaves: Arrange the leaves in ascending order for each stem.

Here’s the stem-and-leaf plot:

Stem (Tens)Leaves (Ones)
21, 2, 3, 8
30, 0, 0, 3, 3, 4, 4, 6, 6
40, 1, 1, 1, 1, 2, 3, 6, 7, 7
52, 2, 3, 5, 6, 9
61, 1, 2, 5, 6, 7

Part a: Interpret the Plot

  • The stems represent the tens place of the ages.
  • The leaves represent the ones place of the ages.
  • Example: The first row (21,2,3,82 \, | \, 1, 2, 3, 8) corresponds to ages 21,22,23,2821, 22, 23, 28.

Part b: Discuss Data Trends

  1. Range: The smallest age is 2121 (stem 2, leaf 1), and the largest is 6767 (stem 6, leaf 7).

    • Range: 6721=4667 - 21 = 46.
  2. Mode: The most frequently occurring age is 4141 (stem 4, leaf 1 appears four times).

  3. Shape: The data appears to be slightly right-skewed, with a higher concentration of values in the lower range of ages (20s and 30s).

Would you like further statistical analysis of this data or help with other interpretations?

Follow-Up Questions:

  1. What other visualizations could help interpret this data (e.g., histogram)?
  2. How can we use this data to analyze the university's faculty age distribution?
  3. Can you calculate measures of central tendency (mean, median, mode)?
  4. How does the stem-and-leaf plot compare to a box plot for summarizing this data?
  5. What insights might this distribution provide about faculty planning?

Tip:

When constructing a stem-and-leaf plot, always ensure leaves are arranged in ascending order to maintain clarity and accuracy.

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Math Problem Analysis

Mathematical Concepts

Data Representation
Stem-and-Leaf Plot
Descriptive Statistics

Formulas

Range = Max - Min

Theorems

No specific theorems; focuses on data visualization and summarization techniques.

Suitable Grade Level

Grades 6-9