Question

1. _, 73, _, 74, _, 81, _, 96

five-number summary:
71 ~ 73 ~ 77 ~ 83 ~ 96

Explanation:

The five - number summary consists of the minimum, the first quartile ($Q_1$), the median ($Q_2$), the third quartile ($Q_3$), and the maximum, in that order. So we can match each position in the data set with the five - number summary values.

Step 1: Identify the minimum value

The five - number summary starts with the minimum. The first value in the five - number summary is 71, so the first blank (the minimum of the data set) should be 71.

Step 2: Identify the first quartile ($Q_1$)

The second value in the five - number summary is 73, but we already have 73 in the data set. The next blank after 73 is the first quartile? Wait, no. Wait, the five - number summary is min, $Q_1$, median, $Q_3$, max. Let's list the data points: let the data points be $x_1$, 73, $x_2$, 74, $x_3$, 81, $x_4$, 96. There are 8 data points. When $n = 8$ (even), the median is the average of the 4th and 5th values. The five - number summary is 71 (min), 73 ($Q_1$), 77 (median), 83 ($Q_3$), 96 (max).

First, min: the smallest value, so $x_1=71$.

Then, $Q_1$: for $n = 8$, the first quartile is the median of the first 4 values. The first 4 values are 71, 73, $x_2$, 74. The median of these 4 (since $n = 4$, even) is the average of the 2nd and 3rd values. We know $Q_1 = 73$, so $\frac{73 + x_2}{2}=73$, which gives $x_2 = 73$? Wait, no, maybe a better way: the five - number summary's second value is $Q_1 = 73$, and we have 73 as the second data point. Then the median: the median is the average of the 4th and 5th data points. The 4th data point is 74, the 5th is $x_3$. The median is 77, so $\frac{74 + x_3}{2}=77$. Solving for $x_3$: $74+x_3 = 154$, so $x_3=80$? Wait, no, the five - number summary's third value is 77 (median). Wait, maybe I made a mistake. Wait, the five - number summary is min = 71, $Q_1=73$, median = 77, $Q_3 = 83$, max = 96.

The data set has 8 values: positions 1 - 8.

Min (position 1): 71.

$Q_1$: for $n = 8$, $Q_1$ is the median of the first 4 values (positions 1 - 4). The first 4 values: 71, 73, $x_2$, 74. The median of these 4 is $\frac{73 + x_2}{2}=Q_1 = 73$, so $73+x_2=146$, $x_2 = 73$. Wait, but then the first 4 values are 71,73,73,74. The median of these is $\frac{73 + 73}{2}=73$, which matches $Q_1 = 73$.

Median: for $n = 8$, median is the average of the 4th and 5th values. The 4th value is 74, the 5th is $x_3$. So $\frac{74 + x_3}{2}=77$, so $74+x_3=154$, $x_3 = 80$? But the five - number summary's median is 77. Wait, maybe the data set is ordered. Let's order the data set. The max is 96, min is 71. Let's list the known values in order: 71, 73, 74, 81, 96. Wait, no, the original data is _, 73, _, 74, _, 81, _, 96. Let's order them: min (71), then 73, then $x_2$, then 74, then $x_3$, then 81, then $x_4$, then max (96).

Median (middle value for even $n$: average of 4th and 5th). The 4th value is 74, 5th is $x_3$. Median is 77, so $\frac{74 + x_3}{2}=77\Rightarrow x_3 = 77\times2 - 74=154 - 74 = 80$? Wait, but the five - number summary's median is 77. Wait, maybe I messed up the position. Wait, 8 data points: indices 0 - 7 (if we start from 0). Then the median is the average of index 3 and index 4. Index 3 is 74, index 4 is $x_3$. So $\frac{74 + x_3}{2}=77\Rightarrow x_3 = 80$. But the five - number summary's median is 77, so that's correct.

Then $Q_3$: for $n = 8$, $Q_3$ is the median of the last 4 values (indices 4 - 7: $x_3$, 81, $x_4$, 96). The median of these 4 (since $n = 4$, even) is the average of the 2nd and 3rd values (index 5 and index 6). We know $Q_3 = 83$, so $\frac{81 + x_4}{2}=83\Rightarrow81+x_4 = 166\Rightarrow x_4 = 85$? Wait, no, the five - number summary's fourth value is 83 ($Q_3$). Wait, maybe I should just match the five - number summary to the data positions.

The five - number summary is min = 71, $Q_1 = 73$, median = 77, $Q_3 = 83$, max = 96.

The data set has 8 values: [min, _, $Q_1$, _, median, _, $Q_3$, max]? No, let's count the number of values. The given data has blanks: _, 73, _, 74, _, 81, _, 96. So that's 8 values. Let's list them in order:

1. Min: 71 (so first blank is 71)
2. Second value: 73
3. Third value: Let's see, $Q_1$ is the median of the first 4 values. The first 4 values are 71, 73, $x_2$, 74. The median of these 4 (since $n = 4$) is $\frac{73 + x_2}{2}=Q_1 = 73$, so $x_2 = 73$ (third blank is 73)
4. Fourth value: 74
5. Fifth value: The median is the average of the 4th and 5th values. We know median = 77, so $\frac{74 + x_3}{2}=77\Rightarrow x_3 = 80$ (fifth blank is 80)
6. Sixth value: 81
7. Seventh value: $Q_3$ is the median of the last 4 values (values 4 - 7: $x_3$, 81, $x_4$, 96). The median of these 4 is $\frac{81 + x_4}{2}=Q_3 = 83\Rightarrow81+x_4 = 166\Rightarrow x_4 = 85$? Wait, no, the five - number summary's $Q_3$ is 83, and the seventh value is $Q_3$? Wait, maybe I made a mistake in the position. Wait, the five - number summary is min, $Q_1$, median, $Q_3$, max. For $n = 8$, the positions are:

- Min: 1st value
- $Q_1$: median of 1st - 4th values (average of 2nd and 3rd values when $n = 4$)
- Median: average of 4th and 5th values
- $Q_3$: median of 5th - 8th values (average of 6th and 7th values when $n = 4$)
- Max: 8th value

So let's re - assign:

1. Min (1st value): 71 (given in five - number summary)
2. 2nd value: 73 (given)
3. 3rd value: Let the 3rd value be $a$. The first 4 values are 71, 73, $a$, 74. The median of these 4 (since $n = 4$) is $\frac{73 + a}{2}=Q_1 = 73$ (from five - number summary). Solving $\frac{73 + a}{2}=73\Rightarrow73 + a=146\Rightarrow a = 73$
4. 4th value: 74 (given)
5. 5th value: Let the 5th value be $b$. The median is the average of the 4th and 5th values: $\frac{74 + b}{2}=77$ (median from five - number summary). Solving $74 + b=154\Rightarrow b = 80$
6. 6th value: 81 (given)
7. 7th value: Let the 7th value be $c$. The last 4 values are 80, 81, $c$, 96. The median of these 4 (since $n = 4$) is $\frac{81 + c}{2}=Q_3 = 83$ (from five - number summary). Solving $81 + c=166\Rightarrow c = 85$? Wait, but the five - number summary's $Q_3$ is 83, but maybe I made a mistake. Wait, the five - number summary is 71 ~ 73 ~ 77 ~ 83 ~ 96. So the order is min = 71, $Q_1 = 73$, median = 77, $Q_3 = 83$, max = 96.

The data set has 8 elements. Let's list them in order:

1. 71 (min)
2. 73
3. 73 ($Q_1$ is the median of first 4, so (73 + 73)/2 = 73)
4. 74
5. 80 (since median is (74 + 80)/2 = 77)
6. 81
7. 85 (since $Q_3$ is the median of last 4: (81 + 85)/2 = 83)
8. 96 (max)

Wait, but let's check the five - number summary:

- Min: 71 (correct)
- $Q_1$: median of first 4 (71,73,73,74) is (73 + 73)/2 = 73 (correct)
- Median: median of all 8 values. Since $n = 8$, median is (4th + 5th)/2=(74 + 80)/2 = 77 (correct)
- $Q_3$: median of last 4 (80,81,85,96) is (81 + 85)/2 = 83 (correct)
- Max: 96 (correct)

So the blanks are filled as follows:

First blank (1st value): 71

Second blank (3rd value): 73

Third blank (5th value): 80

Fourth blank (7th value): 85

Answer:

The filled sequence is $\boldsymbol{71}, 73, \boldsymbol{73}, 74, \boldsymbol{80}, 81, \boldsymbol{85}, 96$