Question

medical marijuana 2015 sales tax: the data show the amount of sales tax paid in a county in colorado. find the first and third quartiles for the data.
month\tsales tax\tmonth\tsales tax
january\t363,946\tjuly\t518,706
february\t358,807\taugust\t554,113
march\t418,590\tseptember\t506,979
april\t266,266\toctober\t341,851
may\t399,810\tnovember\t349,848
june\t453,062\tdecember\t532,061
send data to excel
your answer
part: 0 / 2
part 1 of 2
first quartile ( q_1 ) is

Explanation:

Step1: Order the data

First, we list all the sales tax values and sort them in ascending order:
\( 266266, 341851, 349848, 358807, 363946, 399810, 418590, 453062, 506979, 518706, 532061, 554113 \)
There are \( n = 12 \) data points.

Step2: Find the position of quartiles

The formula for the position of the \( i \)-th quartile is \( Q_i \) position \( = \frac{i(n + 1)}{4} \) for \( i = 1, 3 \) (for quartiles).

For the first quartile (\( Q_1 \)), \( i = 1 \):
\( \text{Position of } Q_1 = \frac{1\times(12 + 1)}{4}=\frac{13}{4} = 3.25 \)

For the third quartile (\( Q_3 \)), \( i = 3 \):
\( \text{Position of } Q_3 = \frac{3\times(12 + 1)}{4}=\frac{39}{4}=9.75 \)

Step3: Calculate \( Q_1 \)

Since the position of \( Q_1 \) is \( 3.25 \), we take the 3rd value and add 0.25 times the difference between the 4th and 3rd values.
The 3rd value is \( 349848 \), the 4th value is \( 358807 \).
\( Q_1=349848 + 0.25\times(358807 - 349848) \)
\( = 349848+0.25\times8959 \)
\( = 349848 + 2239.75=352087.75 \)? Wait, no, maybe we use the method for discrete data with \( n \) even. Wait, another method: for \( n \) data points, the first quartile is the median of the lower half, and the third quartile is the median of the upper half.

The lower half (first 6 data points): \( 266266, 341851, 349848, 358807, 363946, 399810 \)
The median of the lower half (for \( n = 6 \), median is the average of the 3rd and 4th values)
Median of lower half \(=\frac{349848 + 358807}{2}=\frac{708655}{2}=354327.5 \)? Wait, maybe I made a mistake in the first method. Wait, let's check the data again.

Wait the data points are:

January: 363,946

February: 358,807

March: 418,590

April: 266,266

May: 399,810

June: 453,062

July: 518,706

August: 554,113

September: 506,979

October: 341,851

November: 349,848

December: 532,061

Let's sort them correctly:

266266 (April), 341851 (October), 349848 (November), 358807 (February), 363946 (January), 399810 (May), 418590 (March), 453062 (June), 506979 (September), 518706 (July), 532061 (December), 554113 (August)

Now, \( n = 12 \), so the lower half is the first 6 values: 266266, 341851, 349848, 358807, 363946, 399810

The upper half is the last 6 values: 418590, 453062, 506979, 518706, 532061, 554113

The first quartile \( Q_1 \) is the median of the lower half. For a set with 6 values, the median is the average of the 3rd and 4th values.

3rd value in lower half: 349848

4th value in lower half: 358807

\( Q_1=\frac{349848 + 358807}{2}=\frac{708655}{2}=354327.5 \)? Wait, but maybe the problem expects using the percentile formula \( P_k = \text{value at position } \lceil \frac{k(n)}{100} ceil \) or another method. Wait, another approach: for \( n = 12 \), the position of \( Q_1 \) is \( \frac{n + 1}{4}=\frac{13}{4}=3.25 \), so we take the 3rd value plus 0.25 times the difference between the 4th and 3rd.

3rd value: 349848

4th value: 358807

Difference: 358807 - 349848 = 8959

0.25*8959 = 2239.75

So \( Q_1 = 349848 + 2239.75 = 352087.75 \). But maybe the data is considered as 12 points, and we can also use the method where \( Q_1 \) is the median of the first 6, but let's check the third quartile.

For \( Q_3 \), position is \( \frac{3(n + 1)}{4}=\frac{39}{4}=9.75 \), so 9th value plus 0.75 times the difference between 10th and 9th.

9th value: 506979

10th value: 518706

Difference: 518706 - 506979 = 11727

0.75*11727 = 8795.25

\( Q_3 = 506979 + 8795.25 = 515774.25 \)? Wait, this is confusing. Wait, maybe the problem is using the method where we split the data into two halves without including the median (since n is even, there is no single median). Wait, n = 12, so lower half is first 6, upper half is last 6.

Median of lower half (Q1) is average of 3rd and 4th in lower half: (349848 + 358807)/2 = 354327.5

Median of upper half (Q3) is average of 3rd and 4th in upper half: (506979 + 518706)/2 = (1025685)/2 = 512842.5

But maybe the original data was misread. Wait let's list all sales tax values again:

April: 266,266

October: 341,851

November: 349,848

February: 358,807

January: 363,946

May: 399,810

March: 418,590

June: 453,062

September: 506,979

July: 518,706

December: 532,061

August: 554,113

Now, sorted: 266266, 341851, 349848, 358807, 363946, 399810, 418590, 453062, 506979, 518706, 532061, 554113

Number of data points \( n = 12 \)

The formula for quartiles in a data set with \( n \) observations:

- First quartile (\( Q_1 \)): the value at the \( \frac{n + 1}{4} \)-th position (if not integer, interpolate)

- Third quartile (\( Q_3 \)): the value at the \( \frac{3(n + 1)}{4} \)-th position (if not integer, interpolate)

So for \( n = 12 \):

\( Q_1 \) position: \( \frac{12 + 1}{4}=3.25 \)

So we take the 3rd value (349848) and add 0.25 times the difference between the 4th (358807) and 3rd (349848) values.

\( Q_1 = 349848 + 0.25\times(358807 - 349848) \)

\( = 349848 + 0.25\times8959 \)

\( = 349848 + 2239.75 = 352087.75 \)

\( Q_3 \) position: \( \frac{3\times(12 + 1)}{4}=9.75 \)

Take the 9th value (506979) and add 0.75 times the difference between the 10th (518706) and 9th (506979) values.

\( Q_3 = 506979 + 0.75\times(518706 - 506979) \)

\( = 506979 + 0.75\times11727 \)

\( = 506979 + 8795.25 = 515774.25 \)

But maybe the problem is using the method where we use \( n \) instead of \( n + 1 \) in the formula. Let's try that.

For \( Q_1 \), position \( = \frac{n}{4}=\frac{12}{4}=3 \), so the 3rd value: 349848

For \( Q_3 \), position \( = \frac{3n}{4}=\frac{36}{4}=9 \), so the 9th value: 506979

Ah, this is a common method where we use \( \frac{n}{4} \) and \( \frac{3n}{4} \) for quartiles, especially when \( n \) is divisible by 4. Since \( n = 12 \), which is divisible by 4 (12/4 = 3, 3*12/4 = 9), so \( Q_1 \) is the 3rd value, \( Q_3 \) is the 9th value.

Let's check the sorted data:

1: 266266

2: 341851

3: 349848

4: 358807

5: 363946

6: 399810

7: 418590

8: 453062

9: 506979

10: 518706

11: 532061

12: 554113

So the 3rd value is 349848, the 9th value is 506979. This is a valid method (using \( \frac{n}{4} \) and \( \frac{3n}{4} \) as positions, taking the value at that position when \( n \) is divisible by 4). So probably the problem expects this method.

So \( Q_1 = 349848 \), \( Q_3 = 506979 \)

Answer:

First quartile \( Q_1 = 349848 \), Third quartile \( Q_3 = 506979 \)