Question

here is a data set:
68 75 64 72 73
65 72 78 69 55
73 56 60 57 63
60 60 66 64 64
you are examining the data with an expanded stem-and-leaf plot. here is the start of the plot:
5# | 567
6# | 0003444
6# | 5689
7# |
7# |
what should be entered in the second to last row of this table?
ans =
enter an integer or decimal number more..
submit question

Explanation:

Step1: Identify the stem for the second to last row

The stem - and - leaf plot has stems like 5#, 6#, 7#. The second to last row is the row with stem 7# (since the last row would be another 7#? Wait, let's list the data. The data set is: 68, 75, 64, 72, 73, 65, 72, 78, 69, 55, 73, 56, 60, 57, 63, 60, 66, 64, 60, 64. Wait, no, let's re - extract the data from the table. The table has 4 columns and 5 rows? Wait the table is:

Row 1: 68, 75, 64, 72, 73

Row 2: 65, 72, 78, 69

Row 3: 55, 56, 60, 57, 63

Row 4: 60, 66, 64, 60

Row 5: 64

Wait, maybe better to list all data points: Let's collect all numbers: 68, 75, 64, 72, 73, 65, 72, 78, 69, 55, 73, 56, 60, 57, 63, 60, 66, 64, 60, 64.

Now, let's categorize by the tens place (stem):

- Stem 5 (5#): 55, 56, 57. The leaves are 5, 6, 7 (matches 5# | 567)
- Stem 6 (6#): Let's list numbers with tens digit 6: 68, 64, 65, 69, 60, 63, 60, 66, 64, 60, 64. Wait, let's sort them: 60, 60, 60, 63, 64, 64, 64, 65, 68, 69, 66? Wait no, 60, 60, 60, 63, 64, 64, 64, 65, 66, 68, 69. Wait the given stem - and - leaf for 6# is 0003444 and 5689? Wait 6# | 0003444 (leaves 0,0,0,3,4,4,4) and 6# | 5689 (leaves 5,6,8,9). Let's check: 60 (leaf 0), 60 (0), 60 (0), 63 (3), 64 (4), 64 (4), 64 (4) for the first part. Then 65 (5), 66 (6), 68 (8), 69 (9) for the second part. That matches.

Now stem 7 (7#): Numbers with tens digit 7: 75, 72, 73, 72, 78, 73. Let's sort them: 72, 72, 73, 73, 75, 78.

The stem - and - leaf plot has two rows for 7#? Wait the question is about the second to last row. The rows are 5#, 6# (two rows), 7# (two rows). Wait the given plot has:

5# | 567

6# | 0003444

6# | 5689

7# |? (second to last row)

7# |? (last row)

Wait, let's take the numbers with tens digit 7: 72, 72, 73, 73, 75, 78. Let's split them. Let's see the first part of 7# (second to last row) and the last part (last row). Let's sort the numbers: 72, 72, 73, 73, 75, 78.

Let's split into two groups. Let's see the leaves. The first group (second to last row) and the second group (last row). Let's take the first three or four? Wait, 72, 72, 73, 73 are the first four? Wait 72 (leaf 2), 72 (2), 73 (3), 73 (3). Then 75 (5), 78 (8) for the last row. Wait no, maybe the stem - and - leaf for 7#: let's list the leaves. The numbers are 72,72,73,73,75,78. So the leaves are 2,2,3,3,5,8.

If we split into two rows for 7#: the second to last row (first 7# row) and the last row (second 7# row). Let's sort the leaves: 2,2,3,3,5,8. Let's split into two groups. Let's take the first four? No, maybe two groups of three? 2,2,3 and 3,5,8? No. Wait, maybe the first row (second to last) has leaves 2,2,3,3 and the last row has 5,8? Wait no, 72,72,73,73 are four numbers, 75 and 78 are two. Wait the stem - and - leaf plot for 7#: the second to last row (first 7# row) should have the leaves for the numbers 72,72,73,73. Wait, 72 is 7|2, 72 is 7|2, 73 is 7|3, 73 is 7|3. So the leaves are 2,2,3,3. Wait but let's check the data again. Wait the data points with tens digit 7: 75 (7|5), 72 (7|2), 73 (7|3), 72 (7|2), 78 (7|8), 73 (7|3). So that's six numbers: 72,72,73,73,75,78.

So if we have two rows for 7#:

First 7# row (second to last row): let's take the first four numbers: 72,72,73,73. So leaves are 2,2,3,3. Wait but maybe sorted leaves: 2,2,3,3.

Wait the problem is asking for the second to last row of the stem - and - leaf table. The stem is 7#, so we need to find the leaves for the numbers with tens digit 7, in the second to last row.

Wait let's re - list all data points correctly. Let's extract from the table:

The table is:

Column 1: 68, 65, 55, 60, 64

Column 2: 75, 72, 56, 66, (empty?)

Column 3: 64, 78, 60, 64, (empty?)

Column 4: 72, 69, 57, 60, (empty?)

Column 5: 73, (empty), 63, (empty), (empty)

Wait maybe the data set is: 68, 75, 64, 72, 73, 65, 72, 78, 69, 55, 56, 60, 57, 63, 60, 66, 64, 60, 64. Let's count: 19 numbers? Wait no, maybe I miscounted. Let's list all:

From the table:

Row 1 (top row): 68, 75, 64, 72, 73

Row 2: 65, 72, 78, 69

Row 3: 55, 56, 60, 57, 63

Row 4: 60, 66, 64, 60

Row 5: 64

So total numbers: 5 + 4+5 + 4+1 = 19. Now, numbers with tens digit 7: 75,72,73,72,78,73. That's 6 numbers (from row 1: 75,72,73; row 2:72,78; and wait row 1 has 75,72,73; row 2 has 72,78; is there another? Wait row 1: 68,75,64,72,73 (three numbers with tens digit 7:75,72,73); row 2:65,72,78,69 (two numbers:72,78); so total 3 + 2=5? Wait I missed one. Wait row 1: 73 (third number), row 2:73? No, row 1 has 73, row 2 has 72,78. Wait maybe row 1: 68,75,64,72,73 (5 numbers), row 2:65,72,78,69 (4 numbers), so 75,72,73,72,78: that's 5 numbers? Wait 75,72,73,72,78. Then where is the sixth? Oh, maybe row 1: 73 (fifth number), row 2: no, maybe I made a mistake. Wait the stem - and - leaf for 7#: let's take the numbers with tens digit 7: 72,72,73,73,75,78. Let's assume there are six numbers. So sorted: 72,72,73,73,75,78.

Now, the stem - and - leaf plot has 5#, 6# (two rows), 7# (two rows). So the second to last row is the first 7# row, and the last row is the second 7# row.

Let's split the sorted numbers into two groups. Let's take the first four numbers: 72,72,73,73. Their leaves are 2,2,3,3. The last two numbers:75,78 with leaves 5,8.

Wait, but let's check the 6# rows. The first 6# row has leaves 0,0,0,3,4,4,4 (for numbers 60,60,60,63,64,64,64) and the second 6# row has leaves 5,6,8,9 (for 65,66,68,69). So the first row of a stem has the smaller leaves and the second row has the larger leaves.

So for stem 7, the smaller leaves (2,2,3,3) should be in the second to last row (first 7# row) and the larger leaves (5,8) in the last row.

Wait, but let's check the numbers: 72 (leaf 2), 72 (2), 73 (3), 73 (3) are the numbers with leaves 2 and 3. Then 75 (5) and 78 (8) with leaves 5 and 8.

So the second to last row (stem 7#) should have leaves 2,2,3,3. Wait, but the question is asking for what should be entered in the second to last row. Wait, maybe we need to list the leaves in order. So the leaves for the second to last row (7#) are 2,2,3,3. Wait, but let's check the data again. Wait the numbers with tens digit 7 are: 72,72,73,73,75,78. So when we make a stem - and - leaf plot, we sort the leaves. So sorted leaves for 7#: 2,2,3,3,5,8.

If we split into two rows, the first row (second to last) has the first four leaves: 2,2,3,3 and the last row has 5,8.

Wait, but maybe the stem - and - leaf plot is constructed by splitting the stem into two parts, like 70 - 74 and 75 - 79. Let's check:

Numbers between 70 - 74: 72,72,73,73 (since 72,72,73,73 are between 70 and 74). Numbers between 75 - 79:75,78.

Ah! That makes sense. So the stem for 70 - 74 is 7# (with the units digit as leaf) and 75 - 79 is also 7#? Wait no, in an expanded stem - and - leaf plot, we can split the stem into two, so 7 (for 70 - 74) and 7 (for 75 - 79), but with different leaves. Wait, no, the stem is the tens digit, and we can split the stem into two rows: one for leaves 0 - 4 and one for leaves 5 - 9.

Ah! That's the key. In an expanded stem - and - leaf plot, each stem (tens digit) is split into two rows: one for leaves 0 - 4 and one for leaves 5 - 9.

So for stem 7 (tens digit 7):

- Row 1 (leaves 0 - 4): numbers with units digit 0 - 4. The numbers with tens digit 7 and units digit 0 - 4 are 72,72,73,73 (units digits 2,2,3,3 which are between 0 - 4).

- Row 2 (leaves 5 - 9): numbers with units digit 5 - 9. The numbers with tens digit 7 and units digit 5 - 9 are 75 (5), 78 (8) (units digits 5 and 8 which are between 5 - 9).

So the second to last row is the row for stem 7 with leaves 0 - 4, so the leaves are 2,2,3,3. Wait, but the question is asking for what to enter in the second to last row. Wait, maybe we need to list the leaves in order. So sorted leaves for 0 - 4: 2,2,3,3.

Wait, let's verify with the 6# stem. For stem 6 (tens digit 6):

- Leaves 0 - 4: numbers with units digit 0 - 4. The numbers are 60,60,60,63,64,64,64 (units digits 0,0,0,3,4,4,4) which matches 6# | 0003444.

- Leaves 5 - 9: numbers with units digit 5 - 9. The numbers are 65,66,68,69 (units digits 5,6,8,9) which matches 6# | 5689.

Yes! So that's the pattern. So for stem 7:

- Leaves 0 - 4 (second to last row): numbers with tens digit 7 and units digit 0 - 4. The numbers are 72,72,73,73. So the leaves are 2,2,3,3. When we write the stem - and - leaf, we list the leaves in ascending order, so 2,2,3,3.

Wait, but let's check the data again. The numbers with tens digit 7: 75 (7|5), 72 (7|2), 73 (7|3), 72 (7|2), 78 (7|8), 73 (7|3). So units digits: 5,2,3,2,8,3. Sorting the units digits for 0 - 4: 2,2,3,3 (since 2 and 3 are between 0 - 4). The units digits for 5 - 9:5,8.

So the second to last row (stem 7, leaves 0 - 4) has leaves 2,2,3,3. So when we write the stem - and - leaf, it's 7# | 2233? Wait no, we list the leaves in order, so 2,2,3,3. Wait, but maybe we need to count the frequency? No, in stem - and - leaf, we list each leaf.

Wait, the 6# | 0003444 has three 0s, one 3, three 4s. Which matches 60 (0), 60 (0), 60 (0), 63 (3), 64 (4), 64 (4), 64 (4). So for 7# | (leaves 0 - 4), we have two 2s and two 3s, so the

Answer:

Step1: Identify the stem for the second to last row

The stem - and - leaf plot has stems like 5#, 6#, 7#. The second to last row is the row with stem 7# (since the last row would be another 7#? Wait, let's list the data. The data set is: 68, 75, 64, 72, 73, 65, 72, 78, 69, 55, 73, 56, 60, 57, 63, 60, 66, 64, 60, 64. Wait, no, let's re - extract the data from the table. The table has 4 columns and 5 rows? Wait the table is:

Row 1: 68, 75, 64, 72, 73

Row 2: 65, 72, 78, 69

Row 3: 55, 56, 60, 57, 63

Row 4: 60, 66, 64, 60

Row 5: 64

Wait, maybe better to list all data points: Let's collect all numbers: 68, 75, 64, 72, 73, 65, 72, 78, 69, 55, 73, 56, 60, 57, 63, 60, 66, 64, 60, 64.

Now, let's categorize by the tens place (stem):

• Stem 5 (5#): 55, 56, 57. The leaves are 5, 6, 7 (matches 5# | 567)
• Stem 6 (6#): Let's list numbers with tens digit 6: 68, 64, 65, 69, 60, 63, 60, 66, 64, 60, 64. Wait, let's sort them: 60, 60, 60, 63, 64, 64, 64, 65, 68, 69, 66? Wait no, 60, 60, 60, 63, 64, 64, 64, 65, 66, 68, 69. Wait the given stem - and - leaf for 6# is 0003444 and 5689? Wait 6# | 0003444 (leaves 0,0,0,3,4,4,4) and 6# | 5689 (leaves 5,6,8,9). Let's check: 60 (leaf 0), 60 (0), 60 (0), 63 (3), 64 (4), 64 (4), 64 (4) for the first part. Then 65 (5), 66 (6), 68 (8), 69 (9) for the second part. That matches.

Now stem 7 (7#): Numbers with tens digit 7: 75, 72, 73, 72, 78, 73. Let's sort them: 72, 72, 73, 73, 75, 78.

The stem - and - leaf plot has two rows for 7#? Wait the question is about the second to last row. The rows are 5#, 6# (two rows), 7# (two rows). Wait the given plot has:

5# | 567

6# | 0003444

6# | 5689

7# |? (second to last row)

7# |? (last row)

Wait, let's take the numbers with tens digit 7: 72, 72, 73, 73, 75, 78. Let's split them. Let's see the first part of 7# (second to last row) and the last part (last row). Let's sort the numbers: 72, 72, 73, 73, 75, 78.

Let's split into two groups. Let's see the leaves. The first group (second to last row) and the second group (last row). Let's take the first three or four? Wait, 72, 72, 73, 73 are the first four? Wait 72 (leaf 2), 72 (2), 73 (3), 73 (3). Then 75 (5), 78 (8) for the last row. Wait no, maybe the stem - and - leaf for 7#: let's list the leaves. The numbers are 72,72,73,73,75,78. So the leaves are 2,2,3,3,5,8.

If we split into two rows for 7#: the second to last row (first 7# row) and the last row (second 7# row). Let's sort the leaves: 2,2,3,3,5,8. Let's split into two groups. Let's take the first four? No, maybe two groups of three? 2,2,3 and 3,5,8? No. Wait, maybe the first row (second to last) has leaves 2,2,3,3 and the last row has 5,8? Wait no, 72,72,73,73 are four numbers, 75 and 78 are two. Wait the stem - and - leaf plot for 7#: the second to last row (first 7# row) should have the leaves for the numbers 72,72,73,73. Wait, 72 is 7|2, 72 is 7|2, 73 is 7|3, 73 is 7|3. So the leaves are 2,2,3,3. Wait but let's check the data again. Wait the data points with tens digit 7: 75 (7|5), 72 (7|2), 73 (7|3), 72 (7|2), 78 (7|8), 73 (7|3). So that's six numbers: 72,72,73,73,75,78.

So if we have two rows for 7#:

First 7# row (second to last row): let's take the first four numbers: 72,72,73,73. So leaves are 2,2,3,3. Wait but maybe sorted leaves: 2,2,3,3.

Wait the problem is asking for the second to last row of the stem - and - leaf table. The stem is 7#, so we need to find the leaves for the numbers with tens digit 7, in the second to last row.

Wait let's re - list all data points correctly. Let's extract from the table:

The table is:

Column 1: 68, 65, 55, 60, 64

Column 2: 75, 72, 56, 66, (empty?)

Column 3: 64, 78, 60, 64, (empty?)

Column 4: 72, 69, 57, 60, (empty?)

Column 5: 73, (empty), 63, (empty), (empty)

Wait maybe the data set is: 68, 75, 64, 72, 73, 65, 72, 78, 69, 55, 56, 60, 57, 63, 60, 66, 64, 60, 64. Let's count: 19 numbers? Wait no, maybe I miscounted. Let's list all:

From the table:

Row 1 (top row): 68, 75, 64, 72, 73

Row 2: 65, 72, 78, 69

Row 3: 55, 56, 60, 57, 63

Row 4: 60, 66, 64, 60

Row 5: 64

So total numbers: 5 + 4+5 + 4+1 = 19. Now, numbers with tens digit 7: 75,72,73,72,78,73. That's 6 numbers (from row 1: 75,72,73; row 2:72,78; and wait row 1 has 75,72,73; row 2 has 72,78; is there another? Wait row 1: 68,75,64,72,73 (three numbers with tens digit 7:75,72,73); row 2:65,72,78,69 (two numbers:72,78); so total 3 + 2=5? Wait I missed one. Wait row 1: 73 (third number), row 2:73? No, row 1 has 73, row 2 has 72,78. Wait maybe row 1: 68,75,64,72,73 (5 numbers), row 2:65,72,78,69 (4 numbers), so 75,72,73,72,78: that's 5 numbers? Wait 75,72,73,72,78. Then where is the sixth? Oh, maybe row 1: 73 (fifth number), row 2: no, maybe I made a mistake. Wait the stem - and - leaf for 7#: let's take the numbers with tens digit 7: 72,72,73,73,75,78. Let's assume there are six numbers. So sorted: 72,72,73,73,75,78.

Now, the stem - and - leaf plot has 5#, 6# (two rows), 7# (two rows). So the second to last row is the first 7# row, and the last row is the second 7# row.

Let's split the sorted numbers into two groups. Let's take the first four numbers: 72,72,73,73. Their leaves are 2,2,3,3. The last two numbers:75,78 with leaves 5,8.

Wait, but let's check the 6# rows. The first 6# row has leaves 0,0,0,3,4,4,4 (for numbers 60,60,60,63,64,64,64) and the second 6# row has leaves 5,6,8,9 (for 65,66,68,69). So the first row of a stem has the smaller leaves and the second row has the larger leaves.

So for stem 7, the smaller leaves (2,2,3,3) should be in the second to last row (first 7# row) and the larger leaves (5,8) in the last row.

Wait, but let's check the numbers: 72 (leaf 2), 72 (2), 73 (3), 73 (3) are the numbers with leaves 2 and 3. Then 75 (5) and 78 (8) with leaves 5 and 8.

So the second to last row (stem 7#) should have leaves 2,2,3,3. Wait, but the question is asking for what should be entered in the second to last row. Wait, maybe we need to list the leaves in order. So the leaves for the second to last row (7#) are 2,2,3,3. Wait, but let's check the data again. Wait the numbers with tens digit 7 are: 72,72,73,73,75,78. So when we make a stem - and - leaf plot, we sort the leaves. So sorted leaves for 7#: 2,2,3,3,5,8.

If we split into two rows, the first row (second to last) has the first four leaves: 2,2,3,3 and the last row has 5,8.

Wait, but maybe the stem - and - leaf plot is constructed by splitting the stem into two parts, like 70 - 74 and 75 - 79. Let's check:

Numbers between 70 - 74: 72,72,73,73 (since 72,72,73,73 are between 70 and 74). Numbers between 75 - 79:75,78.

Ah! That makes sense. So the stem for 70 - 74 is 7# (with the units digit as leaf) and 75 - 79 is also 7#? Wait no, in an expanded stem - and - leaf plot, we can split the stem into two, so 7 (for 70 - 74) and 7 (for 75 - 79), but with different leaves. Wait, no, the stem is the tens digit, and we can split the stem into two rows: one for leaves 0 - 4 and one for leaves 5 - 9.

Ah! That's the key. In an expanded stem - and - leaf plot, each stem (tens digit) is split into two rows: one for leaves 0 - 4 and one for leaves 5 - 9.

So for stem 7 (tens digit 7):

• Row 1 (leaves 0 - 4): numbers with units digit 0 - 4. The numbers with tens digit 7 and units digit 0 - 4 are 72,72,73,73 (units digits 2,2,3,3 which are between 0 - 4).

• Row 2 (leaves 5 - 9): numbers with units digit 5 - 9. The numbers with tens digit 7 and units digit 5 - 9 are 75 (5), 78 (8) (units digits 5 and 8 which are between 5 - 9).

So the second to last row is the row for stem 7 with leaves 0 - 4, so the leaves are 2,2,3,3. Wait, but the question is asking for what to enter in the second to last row. Wait, maybe we need to list the leaves in order. So sorted leaves for 0 - 4: 2,2,3,3.

Wait, let's verify with the 6# stem. For stem 6 (tens digit 6):

• Leaves 0 - 4: numbers with units digit 0 - 4. The numbers are 60,60,60,63,64,64,64 (units digits 0,0,0,3,4,4,4) which matches 6# | 0003444.

• Leaves 5 - 9: numbers with units digit 5 - 9. The numbers are 65,66,68,69 (units digits 5,6,8,9) which matches 6# | 5689.

Yes! So that's the pattern. So for stem 7:

• Leaves 0 - 4 (second to last row): numbers with tens digit 7 and units digit 0 - 4. The numbers are 72,72,73,73. So the leaves are 2,2,3,3. When we write the stem - and - leaf, we list the leaves in ascending order, so 2,2,3,3.

Wait, but let's check the data again. The numbers with tens digit 7: 75 (7|5), 72 (7|2), 73 (7|3), 72 (7|2), 78 (7|8), 73 (7|3). So units digits: 5,2,3,2,8,3. Sorting the units digits for 0 - 4: 2,2,3,3 (since 2 and 3 are between 0 - 4). The units digits for 5 - 9:5,8.

So the second to last row (stem 7, leaves 0 - 4) has leaves 2,2,3,3. So when we write the stem - and - leaf, it's 7# | 2233? Wait no, we list the leaves in order, so 2,2,3,3. Wait, but maybe we need to count the frequency? No, in stem - and - leaf, we list each leaf.

Wait, the 6# | 0003444 has three 0s, one 3, three 4s. Which matches 60 (0), 60 (0), 60 (0), 63 (3), 64 (4), 64 (4), 64 (4). So for 7# | (leaves 0 - 4), we have two 2s and two 3s, so the