Question

a sample of 20 registered voters was surveyed in which the respondents were asked, \do you think chang, johnson, ohm, or smith is most qualified to be a senator?\ the results of the survey are shown in the table.
smith\tohm\tsmith\tsmith
ohm\tsmith\tsmith\tohm
johnson\tohm\tsmith\tohm
chang\tohm\tchang\tjohnson
ohm\tjohnson\tohm\tjohnson
complete parts (a) and (b) to the right.
(a) determine the mode candidate.
\t○ johnson
\t○ smith
\t○ ohm
\t○ chang

Explanation:

Step1: Count votes for each candidate

- Smith: Let's count. The entries with Smith are: Smith, Smith, Smith, Smith, Smith, Smith. That's 6 times.
- Ohm: Entries with Ohm: Ohm, Ohm, Ohm, Ohm, Ohm, Ohm. Wait, let's count again. The list: Ohm (1), Ohm (from second row: Ohm), Ohm (third row: Ohm), Ohm (fourth row: Ohm), Ohm (fifth row: Ohm), Ohm (last row: Ohm). Wait, no, let's list all:

First column: Smith, Ohm, Johnson, Chang, Ohm

Second column: Ohm, Smith, Ohm, Ohm, Johnson

Third column: Smith, Smith, Smith, Chang, Ohm

Fourth column: Smith, Ohm, Ohm, Johnson, Johnson

Now let's count each:

Smith: First column (1), second column (1), third column (3), fourth column (1). Total: 1+1+3+1 = 6.

Ohm: First column (2), second column (2), third column (1), fourth column (2). Total: 2+2+1+2 = 7? Wait, maybe I made a mistake. Wait let's list all 20 entries:

1. Smith

2. Ohm

3. Smith

4. Smith

5. Ohm

6. Smith

7. Smith

8. Ohm

9. Johnson

10. Ohm

11. Ohm

12. Chang

13. Ohm

14. Chang

15. Johnson

16. Ohm

17. Johnson

18. Smith

19. Ohm

20. Johnson

Wait no, the table has 5 rows and 4 columns, so 5*4=20 entries. Let's list all 20:

Row 1: Smith, Ohm, Smith, Smith

Row 2: Ohm, Smith, Smith, Ohm

Row 3: Johnson, Ohm, Smith, Ohm

Row 4: Chang, Ohm, Chang, Johnson

Row 5: Ohm, Johnson, Ohm, Johnson

Now count each:

Smith: Row1 (3), Row2 (2), Row3 (1) → 3+2+1=6

Ohm: Row1 (1), Row2 (2), Row3 (2), Row4 (1), Row5 (2) → 1+2+2+1+2=8? Wait, no, let's count each entry:

1. Smith

2. Ohm

3. Smith

4. Smith

5. Ohm

6. Smith

7. Smith

8. Ohm

9. Johnson

10. Ohm

11. Ohm

12. Chang

13. Ohm

14. Chang

15. Johnson

16. Ohm

17. Johnson

18. Smith

19. Ohm

20. Johnson

Now let's count:

Smith: 1,3,4,6,7,18 → that's 6 times.

Ohm: 2,5,8,10,11,13,16,19 → 8 times? Wait 2 (Ohm), 5 (Ohm), 8 (Ohm), 10 (Ohm), 11 (Ohm), 13 (Ohm), 16 (Ohm), 19 (Ohm). That's 8 times.

Johnson: 9,15,17,20? Wait row 4: Johnson, row 5: Johnson, row 3: Johnson? Wait row 3: Johnson (first entry), row 4: Johnson (fourth entry), row 5: Johnson (second and fourth entries). So Johnson: 9 (row3), 15 (row4), 17 (row5), 20 (row5). Wait row 5: Johnson (second) and Johnson (fourth). So that's 4 times.

Chang: 12,14 → 2 times.

Wait now I'm confused. Let's do a systematic count:

List all 20 responses:

1. Smith

2. Ohm

3. Smith

4. Smith

5. Ohm

6. Smith

7. Smith

8. Ohm

9. Johnson

10. Ohm

11. Ohm

12. Chang

13. Ohm

14. Chang

15. Johnson

16. Ohm

17. Johnson

18. Smith

19. Ohm

20. Johnson

Now count each:

- Smith: positions 1,3,4,6,7,18 → 6 times.

- Ohm: positions 2,5,8,10,11,13,16,19 → 8 times.

- Johnson: positions 9,15,17,20 → 4 times.

- Chang: positions 12,14 → 2 times.

Wait but the mode is the most frequent value. Wait, but the options are Smith, Ohm, Johnson, Chang. Wait maybe I made a mistake in counting. Let's check again.

Wait the original table:

First column: Smith, Ohm, Johnson, Chang, Ohm → 5 entries.

Second column: Ohm, Smith, Ohm, Ohm, Johnson → 5 entries.

Third column: Smith, Smith, Smith, Chang, Ohm → 5 entries.

Fourth column: Smith, Ohm, Ohm, Johnson, Johnson → 5 entries.

Now count each column:

First column:

Smith: 1

Ohm: 2 (positions 2 and 5)

Johnson:1

Chang:1

Second column:

Ohm:2 (positions 1 and 3,4) Wait second column entries: Ohm, Smith, Ohm, Ohm, Johnson → Ohm appears 3 times (positions 1,3,4), Smith 1, Johnson 1.

Third column:

Smith:3 (positions 1,2,3), Chang 1, Ohm 1.

Fourth column:

Smith:1, Ohm:2 (positions 2,3), Johnson:2 (positions 4,5).

Now total for each:

Smith: first column (1) + second column (1) + third column (3) + fourth column (1) = 1+1+3+1=6.

Ohm: first column (2) + second column (3) + third column (1) + fourth column (2) = 2+3+1+2=8.

Johnson: first column (1) + second column (1) + third column (0) + fourth column (2) = 1+1+0+2=4.

Chang: first column (1) + second column (0) + third column (1) + fourth column (0) = 1+0+1+0=2.

So Ohm has 8, Smith has 6, Johnson 4, Chang 2. Wait but the options include Smith as an option. Did I misread the table?

Wait the original table's data:

First row: Smith, Ohm, Smith, Smith

Second row: Ohm, Smith, Smith, Ohm

Third row: Johnson, Ohm, Smith, Ohm

Fourth row: Chang, Ohm, Chang, Johnson

Fifth row: Ohm, Johnson, Ohm, Johnson

Now let's list all 20:

1. Smith

2. Ohm

3. Smith

4. Smith

5. Ohm

6. Smith

7. Smith

8. Ohm

9. Johnson

10. Ohm

11. Ohm

12. Chang

13. Ohm

14. Chang

15. Johnson

16. Ohm

17. Johnson

18. Smith

19. Ohm

20. Johnson

Wait, now I see that in the fourth column, first entry is Smith (position 18), second is Ohm (19), third is Ohm (16), fourth is Johnson (17), fifth is Johnson (20). Wait no, fourth column has 5 entries: Smith (18), Ohm (19), Ohm (16), Johnson (17), Johnson (20). Wait, maybe the numbering is off. Let's count the number of times each name appears:

Smith: Let's count each occurrence:

- First row: 3 (Smith, Smith, Smith)

- Second row: 1 (Smith)

- Third row: 1 (Smith)

- Fourth row: 0

- Fifth row: 0

Wait no, first row: Smith, Ohm, Smith, Smith → 3 Smiths.

Second row: Ohm, Smith, Smith, Ohm → 2 Smiths.

Third row: Johnson, Ohm, Smith, Ohm → 1 Smith.

Fourth row: Chang, Ohm, Chang, Johnson → 0 Smiths.

Fifth row: Ohm, Johnson, Ohm, Johnson → 0 Smiths.

Total Smith: 3+2+1=6.

Ohm:

First row: 1 (Ohm)

Second row: 2 (Ohm, Ohm)

Third row: 2 (Ohm, Ohm)

Fourth row: 1 (Ohm)

Fifth row: 2 (Ohm, Ohm)

Total Ohm: 1+2+2+1+2=8.

Johnson:

First row: 0

Second row: 1 (Johnson)

Third row: 1 (Johnson)

Fourth row: 1 (Johnson)

Fifth row: 2 (Johnson, Johnson)

Total Johnson: 0+1+1+1+2=5? Wait no, second row: Johnson is the fifth entry? Wait second row has 5 entries: Ohm, Smith, Ohm, Ohm, Johnson → Johnson is 1.

Third row: Johnson, Ohm, Smith, Ohm → Johnson is 1.

Fourth row: Chang, Ohm, Chang, Johnson → Johnson is 1.

Fifth row: Ohm, Johnson, Ohm, Johnson → Johnson is 2.

So Johnson: 1+1+1+2=5.

Chang:

First row: 0

Second row: 0

Third row: 0

Fourth row: 2 (Chang, Chang)

Fifth row: 0

Total Chang: 2.

Ohm:

First row: 1

Second row: 2

Third row: 2

Fourth row: 1

Fifth row: 2

Total: 1+2+2+1+2=8.

Wait, now I'm really confused. But the options are Smith, Ohm, Johnson, Chang. The mode is the most frequent. If Ohm has 8, Smith 6, Johnson 5, Chang 2, then Ohm is the mode. But the options include Smith as an option. Maybe I made a mistake in counting. Let's check the original problem again.

Wait the problem says "a sample of 20 registered voters". Let's count the number of entries:

First column: 5 (Smith, Ohm, Johnson, Chang, Ohm)

Second column: 5 (Ohm, Smith, Ohm, Ohm, Johnson)

Third column: 5 (Smith, Smith, Smith, Chang, Ohm)

Fourth column: 5 (Smith, Ohm, Ohm, Johnson, Johnson)

Now let's count each:

Smith:

First column: 1

Second column: 1

Third column: 3

Fourth column: 1

Total: 1+1+3+1=6.

Ohm:

First column: 2

Second column: 3

Third column: 1

Fourth column: 2

Total: 2+3+1+2=8.

Johnson:

First column: 1

Second column: 1

Third column: 0

Fourth column: 2

Total: 1+1+0+2=4.

Chang:

First column: 1

Second column: 0

Third column: 1

Fourth column: 0

Total: 1+0+1+0=2.

So Ohm has 8, Smith 6, Johnson 4, Chang 2. So the mode should be Ohm? But the options include Smith. Wait maybe the table was presented differently. Wait the original problem's table:

First column: Smith, Ohm, Johnson, Chang, Ohm

Second column: Ohm, Smith, Ohm, Ohm, Johnson

Third column: Smith, Smith, Smith, Chang, Ohm

Fourth column: Smith, Ohm, Ohm, Johnson, Johnson

Wait, fourth column first entry: Smith, second: Ohm, third: Ohm, fourth: Johnson, fifth: Johnson.

Now let's count Smith again:

First column: 1

Second column: 1

Third column: 3

Fourth column: 1

Total: 6.

Ohm:

First column: 2

Second column: 3 (Ohm, Ohm, Ohm)

Third column: 1

Fourth column: 2 (Ohm, Ohm)

Total: 2+3+1+2=8.

Johnson:

First column: 1

Second column: 1

Third column: 0

Fourth column: 2

Total: 4.

Chang:

First column: 1

Second column: 0

Third column: 1

Fourth column: 0

Total: 2.

So Ohm is the most frequent with 8 votes. But the options given are Johnson, Smith, Ohm, Chang. So the answer should be Ohm? Wait but maybe I made a mistake. Wait let's check the number of entries. 5 rows ×4 columns=20 entries. Let's count the total:

Smith: 6, Ohm:8, Johnson:4, Chang:2. 6+8+4+2=20. Correct.

So the mode is the value with the highest frequency, which is Ohm with 8. But wait the options include Ohm as an option (the third option: Ohm). So the answer should be Ohm? But wait maybe the original table was different. Wait the user's table:

First column: Smith, Ohm, Johnson, Chang, Ohm

Second column: Ohm, Smith, Ohm, Ohm, Johnson

Third column: Smith, Smith, Smith, Chang, Ohm

Fourth column: Smith, Ohm, Ohm, Johnson, Johnson

Wait, maybe I miscounted Smith. Let's count Smith again:

First column: 1 (Smith)

Second column: 1 (Smith)

Third column: 3 (Smith, Smith, Smith)

Fourth column: 1 (Smith)

Total: 1+1+3+1=6. Correct.

Ohm:

First column: 2 (Ohm, Ohm)

Second column: 3 (Ohm, Ohm, Ohm)

Third column: 1 (Ohm)

Fourth column: 2 (Ohm, Ohm)

Total: 2+3+1+2=8. Correct.

So the mode is Ohm. But let's check the options. The options are:

- Johnson

- Smith

- Ohm

- Chang

So the correct option is Ohm. Wait but maybe I made a mistake. Wait let's check the number of times Smith appears. Wait in the third column, there are three Smiths (Smith, Smith, Smith), second column has one Smith, first column has one, fourth column has one. So 3+1+1+1=6. Ohm has 8. So Ohm is the mode.

Answer:

Ohm (the third option: Ohm)