Problem 1

Last updated
Save as PDF

Page ID: 275304

Contributor
Analytical Sciences Digital Library

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

One approach to this problem is to collect a sample of pennies (perhaps from your change jar) and to measure the mass of each penny. Open Data Set 1, which is an Excel file containing the masses for 32 pennies. Note that the data consists of one response (the mass of the penny) and one factor (each penny's ID number).

Task 1. Characterize the data quantitatively by calculating the mean and standard deviation. (Use the following link for help with using Excel.) In several sentences, explain the meaning of these two values and what they suggest about this set of data.

Task 2. Characterize the data visually by creating a scatterplot with mass on the y-axis and the ID number on the x-axis. Be sure to scale the axes so that the data occupies most of the available space. In several sentences, explain what information this graph conveys about the data. In what ways do your quantitative and visual characterizations of the data provide similar and/or different information about the data?

Task 3. The visual presentation of the data should strike you as interesting and unusual. Look carefully at your graph of the data. What does it suggest about this sample of pennies? (If you are stuck, try the following hint). Can you think of factor(s) that might explain the variation in the mass of these 32 pennies?

Hint

Where along the y-axis do you find the masses of pennies?

Where along the y-axis do you not find the masses of pennies?

Is this pattern what you might expect for a random sample of pennies?

Task 4. Estimate the uncertainty in measuring the mass of a single penny. For example, if a penny has a mass of 2.512 g, is the uncertainty in its mass 0.1 g, 0.01 g, 0.001 g or 0.0001 g? Compare your estimate for the uncertainty in a penny's mass with your calculated standard deviation. Is this comparison consistent with your conclusions from Task 3? Explain.

After completing these tasks, proceed to Problem 2.