Question

chevrolet 81
nissan 92
toyota 92
ford 95
honda 94
hyundai 80
acura 72
what is the upper and lower quartile of this set of data?
○ lower quartile: 80.5
upper quartile: 93
○ lower quartile: 88
upper quartile: 93
○ lower quartile: 80.5
upper quartile: 88
○ lower quartile: 72
upper quartile: 95

Explanation:

Step1: Order the data

First, we order the data from least to greatest: \(72, 80, 81, 92, 92, 94, 95\) (Wait, original data: Chevrolet 81, Nissan 92, Toyota 92, Ford 95, Honda 94, Hyundai 80, Acura 72. Wait, let's count the number of data points. Wait, the table has Chevrolet, Nissan, Toyota, Ford, Honda, Hyundai, Acura? Wait, the first row: Chevrolet 81, then Nissan 92, Toyota 92, Ford 95, Honda 94, Hyundai 80, Acura 72. Wait, that's 7 data points? Wait, no, maybe I miscounted. Wait, the table: Chevrolet (81), Nissan (92), Toyota (92), Ford (95), Honda (94), Hyundai (80), Acura (72). So 7 data points. Wait, but quartiles: for \(n = 7\), the median is the 4th term. Let's order them: \(72, 80, 81, 92, 92, 94, 95\). So median (Q2) is 92. Then lower half: \(72, 80, 81\) (first 3 terms), upper half: \(92, 94, 95\) (last 3 terms). Wait, no, for odd \(n\), we exclude the median for quartiles? Wait, different methods: some use inclusive, some exclusive. Wait, maybe the data is 9 points? Wait, the original table: maybe I missed a row. Wait, the user's table: Chevrolet 81, Nissan 92, Toyota 92, Ford 95, Honda 94, Hyundai 80, Acura 72. Wait, that's 7. Wait, maybe the problem has 9 data points? Wait, maybe a typo. Wait, let's check the options. The first option: lower quartile 80.5, upper 93. Let's re - order the data correctly. Wait, maybe the data is: 72, 80, 81, 92, 92, 94, 95? No, that's 7. Wait, maybe the original data has more entries. Wait, perhaps the table is: Chevrolet 81, Nissan 92, Toyota 92, Ford 95, Honda 94, Hyundai 80, Acura 72, and maybe two more? Wait, no, the user's image shows: Chevrolet (81), Nissan (92), Toyota (92), Ford (95), Honda (94), Hyundai (80), Acura (72). So 7 data points. Wait, but quartiles: for \(n = 7\), the lower quartile (Q1) is the median of the lower half. Lower half: 72, 80, 81 (3 terms), median is 80.5? Wait, no, 3 terms: median is the 2nd term? Wait, no, for 3 terms, the median is the middle one, which is 80? Wait, no, 72, 80, 81: the middle is 80? Wait, no, 72 (1st), 80 (2nd), 81 (3rd). The median of these three is 80? Wait, no, median of 3 numbers is the 2nd. So 80. But the first option has 80.5. Wait, maybe the data is 8 points? Wait, maybe I missed a data point. Wait, let's assume the data is: 72, 80, 81, 92, 92, 94, 95, and another? Wait, no. Wait, maybe the original data is: 72, 80, 81, 92, 92, 94, 95 (7 points). Wait, but the first option: Q1 = 80.5, Q3 = 93. Let's calculate Q1 and Q3 properly.

Wait, maybe the data is ordered as: 72, 80, 81, 92, 92, 94, 95 (7 points). Wait, no, maybe the data is 9 points. Wait, perhaps the table has 9 rows. Wait, the user's image: maybe the first row is Chevrolet 81, then Nissan 92, Toyota 92, Ford 95, Honda 94, Hyundai 80, Acura 72, and two more? Wait, maybe it's a mistake. Alternatively, let's use the formula for quartiles. For a set of data with \(n\) elements, ordered as \(x_1\leq x_2\leq\cdots\leq x_n\).

If \(n\) is odd, the median is at position \(\frac{n + 1}{2}\). For \(n = 9\), median is at 5th position. But in our case, maybe the data is 9 points. Wait, let's check the options. The first option: Q1 = 80.5, Q3 = 93. Let's assume the data is: 72, 80, 81, 92, 92, 94, 95, and two more? No, this is getting confusing. Wait, let's re - order the given data: 72, 80, 81, 92, 92, 94, 95 (7 points). Wait, no, maybe the original data is: 72, 80, 81, 92, 92, 94, 95 (7 points). Wait, but quartiles:

For \(n = 7\):

- Median (Q2) is at position \(\frac{7+1}{2}=4\), so \(x_4 = 92\).

- Lower half: \(x_1, x_2, x_3\) (72, 80, 81). The median of the lower half (Q1) is the median of these three, which is \(x_2 = 80\)? But the first option has 80.5. Wait, maybe the data is 8 points. Let's assume the data is 8 points: 72, 80, 81, 92, 92, 94, 95, and another? Wait, no. Alternatively, maybe the data is: 72, 80, 81, 92, 92, 94, 95 (7 points), but we use the method of averaging for quartiles when \(n\) is odd. Wait, Q1 is the median of the lower half. Lower half has 3 points: 72, 80, 81. The median of these three is 80. But the first option has 80.5. Wait, maybe the data is 8 points. Let's suppose the data is: 72, 80, 81, 92, 92, 94, 95, and 88? No, the options have 80.5. Wait, maybe the data is ordered as: 72, 80, 81, 92, 92, 94, 95 (7 points). Wait, no, let's calculate Q1 and Q3 using the formula for quartiles:

For a set of data, the lower quartile (Q1) is the value at the \(\frac{n + 1}{4}\)th position, and the upper quartile (Q3) is at the \(\frac{3(n + 1)}{4}\)th position.

For \(n = 7\):

- Q1 position: \(\frac{7 + 1}{4}=2\), so Q1 is the 2nd term, which is 80.

- Q3 position: \(\frac{3(7 + 1)}{4}=6\), so Q3 is the 6th term, which is 94. But this doesn't match the options.

Wait, maybe the data is 9 points. Let's assume the data is: 72, 80, 81, 88, 92, 92, 94, 95, 95 (but no, the options don't have 88 as Q1 with 94 as Q3). Wait, the first option: Q1 = 80.5, Q3 = 93. Let's calculate Q1 and Q3 for 8 data points. Suppose the data is: 72, 80, 81, 92, 92, 94, 95, and another? No, 8 points. Wait, 72, 80, 81, 92, 92, 94, 95, x. But we don't know x. Alternatively, maybe the data is ordered as: 72, 80, 81, 92, 92, 94, 95 (7 points), but the first option's Q1 is 80.5, which is the average of 80 and 81. Ah! Maybe the lower half is considered as 72, 80, 81, 92 (no, median is 92). Wait, no, maybe the data is 8 points. Wait, let's take the first option: Q1 = 80.5 (average of 80 and 81), Q3 = 93 (average of 92 and 94). Let's see: if the data is 8 points, ordered as: 72, 80, 81, 92, 92, 94, 95, y. But no, maybe the original data has 8 points. Wait, the given data in the table: Chevrolet 81, Nissan 92, Toyota 92, Ford 95, Honda 94, Hyundai 80, Acura 72. That's 7 points. Wait, maybe a typo, and there's an 8th point. Let's assume the data is 72, 80, 81, 92, 92, 94, 95 (7 points). But to get Q1 = 80.5, we need to take the average of 80 and 81, which would be if the lower half has 4 points? Wait, no. Wait, maybe the data is 9 points. Wait, I think I made a mistake. Let's re - order the given data: 72, 80, 81, 92, 92, 94, 95 (7 points). Wait, the first option: lower quartile 80.5, upper 93. Let's calculate Q1 and Q3 as follows:

If we consider the data as 8 points (adding a duplicate or something), but no. Wait, maybe the correct approach is:

1. Order the data: 72, 80, 81, 92, 92, 94, 95 (7 values).

2. Find the median (Q2): the 4th value, which is 92.

3. Lower half: 72, 80, 81 (values below Q2).

4. Upper half: 92, 94, 95 (values above Q2). Wait, but 92 is the median, so some methods include it in both halves.

If we include the median in both halves (inclusive method) for \(n = 7\):

- Lower half: 72, 80, 81, 92 (4 values)

- Upper half: 92, 92, 94, 95 (4 values)

Then Q1 is the median of the lower half: average of 80 and 81: \(\frac{80 + 81}{2}=80.5\)

Q3 is the median of the upper half: average of 92 and 94: \(\frac{92+94}{2}=93\)

Ah! That's the first option. So we use the inclusive method, where we include the median in both the lower and upper halves when \(n\) is odd.

So:

Step1: Order the data

Order the data values: \(72, 80, 81, 92, 92, 94, 95\)

Step2: Find the median (Q2)

Since \(n = 7\) (odd), the median is the \(\frac{7 + 1}{2}=4\)th value, which is \(92\).

Step3: Determine lower and upper halves (inclusive)

- Lower half (includes median): \(72, 80, 81, 92\)
- Upper half (includes median): \(92, 92, 94, 95\)

Step4: Calculate Q1 (median of lower half)

The lower half has 4 values. The median of 4 values is the average of the 2nd and 3rd values. So \(Q1=\frac{80 + 81}{2}=80.5\)

Step5: Calculate Q3 (median of upper half)

The upper half has 4 values. The median of 4 values is the average of the 2nd and 3rd values. So \(Q3=\frac{92+94}{2}=93\)

Answer:

lower quartile: 80.5
upper quartile: 93