A person weighed 15 pennies on a balance and recorded the...

A person weighed 15 pennies on a balance and recorded the following masses: $$ \begin{array}{|lll|} \hline 3.112 \mathrm{~g} & 3.109 \mathrm{~g} & 3.059 \mathrm{~g} \\ 2.467 \mathrm{~g} & 3.079 \mathrm{~g} & 2.518 \mathrm{~g} \\ 3.129 \mathrm{~g} & 2.545 \mathrm{~g} & 3.050 \mathrm{~g} \\ 3.053 \mathrm{~g} & 3.054 \mathrm{~g} & 3.072 \mathrm{~g} \\ 3.081 \mathrm{~g} & 3.131 \mathrm{~g} & 3.064 \mathrm{~g} \\ \hline \end{array} $$ Curious about the results, he looked at the dates on each penny. Two of the light pennies were minted in 1983 and one in 1982 . The dates on the 12 heavier pennies ranged from 1970 to $1982 .$ Two of the 12 heavier pennies were minted in 1982 . a. Do you think the Bureau of the Mint changed the way it made pennies? Explain. b. The person calculated the average mass of the 12 heavy pennies. He expressed this average as $3.0828 \mathrm{~g} \pm 0.0482 \mathrm{~g} .$ What is wrong with the numbers in this result, and how should the value be expressed?

A person weighed 15 pennies on a balance and recorded the following masses:
$$
\begin{array}{|lll|}
\hline 3.112 \mathrm{~g} & 3.109 \mathrm{~g} & 3.059 \mathrm{~g} \\
2.467 \mathrm{~g} & 3.079 \mathrm{~g} & 2.518 \mathrm{~g} \\
3.129 \mathrm{~g} & 2.545 \mathrm{~g} & 3.050 \mathrm{~g} \\
3.053 \mathrm{~g} & 3.054 \mathrm{~g} & 3.072 \mathrm{~g} \\
3.081 \mathrm{~g} & 3.131 \mathrm{~g} & 3.064 \mathrm{~g} \\
\hline
\end{array}
$$
Curious about the results, he looked at the dates on each penny. Two of the light pennies were minted in 1983 and one in 1982 . The dates on the 12 heavier pennies ranged from 1970 to $1982 .$ Two of the 12 heavier pennies were minted in 1982 .
a. Do you think the Bureau of the Mint changed the way it made pennies? Explain.
b. The person calculated the average mass of the 12 heavy pennies. He expressed this average as $3.0828 \mathrm{~g} \pm 0.0482 \mathrm{~g} .$ What is wrong with the numbers in this result, and how should the value be expressed?

Key Concepts

Manufacturing Process Consistency

This concept involves comparing different sets of measurements to determine whether there are systematic differences that might indicate a deliberate change in the manufacturing process. It is a critical part of quality control in production since a shift in the weight or other properties of manufactured items may signal changes in material composition, molding techniques, or quality standards.

Statistical Analysis of Sample Data

This concept covers the use of statistical measures such as the average (mean), variance, and standard deviation to summarize and interpret a set of measurements. It includes understanding how to separate distinct groups within data, deciding if observed differences are statistically significant, and drawing conclusions about overall process performance.

Measurement Uncertainty and Reporting Precision

Measurement uncertainty addresses the doubt associated with any measurement process and is reflected in the reported error. Reporting precision involves using appropriate significant figures consistent with the precision of the instruments used. Overstating the precision of an average or its uncertainty by including too many digits can be misleading because it implies a level of accuracy not supported by the measurement system.

Transcript

00:01 So this question is asking us to make an observation about a data set.

00:05 So this person weighed 15 pennies of varying masses and he's had seen pierre shape had seen that most of the heavier pennies were from 1982 um, or before and some of the lighter pennies were minted in 1982 and next 83.

00:23 So they ask us, um, is it possible that the bureau of the mittens change the way it made tenis? um, and i think that's an accurate conclusion to come to based on the data.

00:35 Right? so, um, pennies.

00:39 Actually, if you go to question 121 writes reviews, read the text in the next question, they explain to you what happened in 1982.

00:47 So sometime in 1982 the mints just decided to make pennies with less copper and more zinc.

00:56 Right, so copper is a more dense metal than zinc.

00:59 So by changing the composition of the penny overall, the penny became much lighter and actually a lot more cheap to produce.

01:07 Um, so yeah, it's definitely possible.

01:10 The bureau of the mints started making their pennies differently in 1982 and that's exactly what was the case in historical context, right? so the issue of this problem comes from part b, where the person who had weighed all the pennies took the average and he had recorded the average to be 3.8 to 8 plus or minus 0.482 grands.

01:37 And the question is asking us what is wrong with numbers and this results, and how should the value be expressed? well, if you will get, um, the question, you'll notice that the pennies the mass of the pennies on lee goes to the thousands place, right? so some of the mass is that it recorded our 3.112 grams.

01:55 Ah, there's a 2.545 grams, right.

01:58 They all go to the thousands place.

02:01 You'll notice the answer, um, stops at the 10/1000 place...

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