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	sales tax	month	sales tax
february	358,807	august	554,113
march	418,590	september	506,979
april	266,266	october	341,851
may	399,810	november	349,848
june	453,062	december	532,061

send data to excel

part: 0 / 2

part 1 of 2

first quartile ( q_1 ) is (square).

Explanation:

Step1: Order the data

First, we list all the sales tax values and sort them in ascending order:
\( 266266, 341851, 349848, 358807, 363946, 399810, 418590, 453062, 506979, 518706, 532061, 554113 \)

Step2: Find the median (to split data)

The number of data points \( n = 12 \) (even). The median is the average of the \( \frac{n}{2} = 6 \)-th and \( \frac{n}{2}+1 = 7 \)-th values.
6th value: \( 399810 \), 7th value: \( 418590 \)
Median \( = \frac{399810 + 418590}{2} = 409200 \)

The data is split into two halves:
Lower half (first 6 values): \( 266266, 341851, 349848, 358807, 363946, 399810 \)
Upper half (last 6 values): \( 418590, 453062, 506979, 518706, 532061, 554113 \)

Step3: Find \( Q_1 \) (median of lower half)

For the lower half, \( n_{lower} = 6 \) (even). The median of the lower half is the average of the \( \frac{6}{2} = 3 \)-rd and \( \frac{6}{2}+1 = 4 \)-th values.
3rd value: \( 349848 \), 4th value: \( 358807 \)
But wait, actually, for quartiles, when \( n \) is even, sometimes we use the method where \( Q_1 \) is the median of the first \( \frac{n}{2} \) values (using the inclusive method or exclusive method). Wait, another common method for quartiles when \( n \) is even:

The position of \( Q_1 \) is \( \frac{n + 1}{4} = \frac{12 + 1}{4} = 3.25 \) (using the formula \( i = \frac{p(n + 1)}{100} \), for \( p = 25 \), \( i = 3.25 \))

So \( Q_1 = \text{value at } 3\text{rd} + 0.25\times(\text{value at } 4\text{th} - \text{value at } 3\text{rd}) \)
Value at 3rd: \( 349848 \), Value at 4th: \( 358807 \)
\( Q_1 = 349848 + 0.25\times(358807 - 349848) = 349848 + 0.25\times8959 = 349848 + 2239.75 = 352087.75 \)? Wait, no, maybe I made a mistake in the initial sorting. Wait, let's re - sort the data correctly:

Wait the sales tax values:

January: 363946

February: 358807

March: 418590

April: 266266

May: 399810

June: 453062

July: 518706

August: 554113

September: 506979

October: 341851

November: 349848

December: 532061

Now let's sort them in ascending order:

April: 266266

October: 341851

November: 349848

February: 358807

January: 363946

May: 399810

March: 418590

June: 453062

September: 506979

July: 518706

December: 532061

August: 554113

Now \( n = 12 \)

Position of \( Q_1 \): \( \frac{n + 1}{4}=\frac{12 + 1}{4}=3.25 \)

So the 3rd value is 349848, the 4th value is 358807

\( Q_1=349848 + 0.25\times(358807 - 349848)=349848+0.25\times8959 = 349848 + 2239.75 = 352087.75 \). But this contradicts the earlier thought. Wait, maybe we use the median of the lower half (inclusive). The lower half is the first 6 values: 266266, 341851, 349848, 358807, 363946, 399810. The median of these 6 values is the average of the 3rd and 4th values. 3rd: 349848, 4th: 358807. So \( Q_1=\frac{349848 + 358807}{2}=\frac{708655}{2}=354327.5 \). No, this is confusing. Wait, maybe the problem expects the use of the "tukey's hinges" method or the simple method where for \( n \) even, we split into two halves.

Wait, another approach: when \( n = 12 \), the first quartile is the median of the first 6 observations, and the third quartile is the median of the last 6 observations.

First 6 observations (sorted): 266266, 341851, 349848, 358807, 363946, 399810

Median of first 6 ( \( Q_1 \)): average of 3rd and 4th term. 3rd term: 349848, 4th term: 358807. \( Q_1=\frac{349848 + 358807}{2}=354327.5 \). But this doesn't seem right. Wait, maybe I sorted the data wrong. Let's check the 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

Yes, that's correct.

Wait, maybe the problem is using the method where \( Q_1 \) is the value at position \( \frac{n}{4} \) when \( n \) is divisible by 4? No, \( n = 12 \), \( \frac{n}{4}=3 \). So the 3rd value is 349,848. And \( Q_3 \) is at position \( \frac{3n}{4}=9 \), the 9th value is 506,979.

Ah, maybe that's the case. Let's check:

For \( n = 12 \), the positions for quartiles:

\( Q_1 \) position: \( \frac{n}{4}=3 \) (1 - based index)

\( Q_3 \) position: \( \frac{3n}{4}=9 \) (1 - based index)

So the 3rd value in the sorted list is 349,848 (November), and the 9th value is 506,979 (September).

Let's verify with the formula for quartiles:

When \( n \) is a multiple of 4, \( Q_1 \) is the value at \( \frac{n}{4} \), \( Q_2 \) at \( \frac{n}{2} \), \( Q_3 \) at \( \frac{3n}{4} \)

Here \( n = 12 \), \( \frac{n}{4}=3 \), \( \frac{3n}{4}=9 \)

So \( Q_1 \) is the 3rd value: 349,848

\( Q_3 \) is the 9th value: 506,979

Answer:

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