Holmes 1
The Age of a Coin
Math 171 – Section 14 – Fall 2014
November 26 th , 2014
Amanda Holmes
I pledge that I have neither given nor received help, nor am I aware of any Honor Code fraction.

Holmes 2
My fifty quarters were collected from rolls of quarters that have been in my possession since I was a child and from spare change. I think that my method may produce bias because the rolled coins were put into the rolls a decade ago meaning that there aren’t any recently minted coins in the rolls.
My fifty pennies were collected from rolls of pennies that have been in my possession since I was a child but the majority came from my spare change. I think that my method may produce bias because the rolled coins were put into the rolls a decade ago meaning that there weren’t any recently minted coins in the rolls.
The project’s population are the quarters and pennies in circulation and the individuals are the fifty quarters and pennies that I have collected for the project. The quantitative variable present in this project is the individual ages of the coins. The qualitative variables are the types of coins and whether the quarters possess an eagle on its tail.
Quarters
Age (years)
1 | 2: represents 12 leaf unit: 1 n: 50
0 | 2
0 | 778899
1 | 0233333334444444
1 | 555556678999
2 | 1334
2 | 589
3 | 00013444
Pennies
Age (years)
1 | 2: represents 12 leaf unit: 1 n: 50
0 | 0000001111112223444
0 | 5666888889999
1 | 00022233444
1 | 5599
2 | 23
2 | 7

Holmes 3
Mean
Quarters
17.78
Standard Deviation 8.1574
Minimum
Q1
Median
Q3
Maximum
2
13
15
24.5
34
Pennies
8
6.737
0
1
8
13
27
After examining the stem plot and histogram for the penny distribution, we can see that there is an extreme right skew and a single peak. There are no outliers and the center of the distribution and the median are equivalent at the coin age of 8. I also found that the middle 50% of the data is between Q1
(1) and the median (8). I think that distribution has an extreme right skew because the majority of my pennies came from my spare change in recent months so the coins would be more likely to have a younger age as compared to the pennies that were rolled. The few older pennies from the rolls caused the right skew.
For the quarters, the distribution has a slight right skew and is single peaked. There are no outliers in the distribution. The center of the distribution and the median are equivalent at the coin age of 15.
The middle 50% of the data is between 1.5 and 9.8. I think the distribution has its shape because most of the quarters came from the rolls, making them older than the individuals. After looking at the stem plot, you can see the majority of the quarters are ten years or older, further illustrating that these coins came from the older rolls as compared to the few quarters that came from spare change.
I do not think the distribution of all coins in circulation is similar to the distribution of my samples because the majority of the individuals in my samples came from rolls of coins that had been collected decades ago. I think that distribution of the coins in circulation would have a more balanced distribution of older and newer minted coins while my samples are skewed because of the older, rolled coins.
Quarters
Pennies
95% Confidence
Interval
(15.51, 20.04)
(6.13, 9.87
Margin Of Error
2.26
1.87
The data for both test were found by using a ZInterval Test.

Holmes 4
1.
If you wanted to estimate the average age of pennies to within one year with 99% confidence, then how large of a sample size would you need to obtain? Use the standard deviation from your sample as your best estimate of the population standard deviation.
For this problem, I will use ( 𝑚. 𝑜. 𝑒. = 𝑍 ( 𝜎
√𝑛
) ) to find n. Here is the data.
Margin of Error = 1
Z* = 2.576
Standard Deviation = 6.737
After plugging in my data, I found that 𝑛 = 301.179
2.
Suppose you want to estimate the proportion of pennies in circulation that are older than 20 years to within 2% with 98% confidence. Use your sample as a pilot study to determine the sample size necessary to achieve your goal. 𝑧∗
For this formula, I will use ( 𝑛 = ( 𝑚.𝑜.𝑒.
)
2 𝑥 𝑝^(1 − 𝑝^) ) to find n. Here is the data.
Z* = 2.326
Margin of error = 0.02
P^ = 0.06
After plugging the data into the formula, I found that 𝑛 = 762.8
.
3.
Suppose I define rare pennies to be those which are old enough to comprise 2% or less of the population. Suppose further that in order to calculate the starting age of rare coins, I plan to use
InvNorm(.98, 𝑥̅ , s ), where 𝑥̅ is your sample mean and s is your sample standard deviation for pennies. Would this be a good technique? Why/why not?

This would not be a good technique because it would require a normal distribution. The current distribution is not normal because of the right skew; therefore, the InvNorm function would not be correct. Also, InvNorm would be targeting 𝝁 and not 𝑥̅ . They are two entirely different pieces of data and thus InvNorm would be unlikely to find 𝑥̅ .
4.
Is the mean age of all quarters in circulation the same as the mean age of all pennies in circulation? You will need to use an appropriate inferential technique to the answer this question.
Make sure that you show (and explain) all work that supports your conclusion. You should also produce a nice graphical representation of the distributions of the ages of your two coin types that makes it easy to compare the two distributions visually as well as discuss the assumptions necessary to run this inference test in your particular situation.
For this problem, I will use the 2SanpletTest. My null hypothesis is that the mean age of all quarters in circulation is equal to the mean age of all pennies in circulation. My alternative hypothesis is that the mean age of all quarters in circulation is not equal to the mean age of all pennies in circulation.

Holmes 5
Here is the distributions of the ages of the quarters and pennies. Again, when looking at the two, you can see the differences in the shape and spread of the distributions.
Quarters
Age (years)
1 | 2: represents 12 leaf unit: 1 n: 50
0 | 2
0 | 778899
1 | 0233333334444444
1 | 555556678999
2 | 1334
2 | 589
3 | 00013444
Pennies
Age (years)
1 | 2: represents 12 leaf unit: 1 n: 50
0 | 0000001111112223444
0 | 5666888889999
1 | 00022233444
1 | 5599
2 | 23
2 | 7
Here is the data based on the information above that will be plugged into the 2SampleTtest.
X
1
= 17.78 X
2
= 8
Sx
1
= 8.1574
N
1
= 50
Sx
2
= 6.737
N
2
= 50
Using the data above, I ran a 2SampleTtest in the calculator and found the following: t = 6.536 p = 0.00000000316
Due to the low p value, we will reject the null hypothesis. There is significant evidence that the mean age of all quarters in circulation is not equal the mean age of all pennies in circulation.
5.
Working under the assumption that coins stay in circulation for thirty years and examining coin production figures from the last thirty years, I hypothesize that the proportion of quarters without an eagle on the “tails” side is 0.62. Use your data to address the validity of my hypothesis using an appropriate inferential technique.

I used the 1PropZtest in the calculator to work out this problem. My P o
= 0.62, X = 23, and n =
50. My null hypothesis is that the proportion of quarters without an eagle on its tail equals 0.62 while my alternative hypothesis is that the proportion of quarters without an eagle on its tail does not equal 0.62. After running the test, z = -2.33 and p = 0.19. Due to the high p value, I reject the alternative hypothesis. There is significant evidence that the proportion of quarters without an eagle on its tail equals 0.62.