Question

below are 36 sorted ages of an acting award winner. find ( p_{80} ) using the method presented in the textbook.
17, 32, 48, 70
19, 32, 53, 72
23, 33, 54, 78
24, 35, 61, 79
26, 37, 62, 79
26, 37, 64, 80
27, 43, 64, 84
27, 46, 68, 88
27, 46, 68, 99
30, 49, 70, ... (remaining data as per image)

Explanation:

Step1: Determine the position of \( P_{80} \)

The formula to find the position of the \( p \)-th percentile is \( i = \frac{p}{100} \times n \), where \( n \) is the number of data points. Here, \( p = 80 \) and \( n = 36 \). So, \( i=\frac{80}{100}\times36 = 28.8 \).

Step2: Interpret the position

Since \( i = 28.8 \) is not an integer, we round up to the next integer, which is \( 29 \). This means we need to find the 29th value in the sorted data set.

Step3: List and count the sorted data

First, let's list out the sorted data (from the given table, we can see the data points are: 17, 19, 23, 24, 26, 26, 27, 27, 27, 30, 32, 32, 33, 35, 37, 37, 43, 46, 46, 48, 49, 53, 54, 61, 62, 64, 64, 64, 66, 66, 70, 70, 72, 78, 79, 79, 80). Wait, let's count them properly. Let's list all 36 data points:

1. 17
2. 19
3. 23
4. 24
5. 26
6. 26
7. 27
8. 27
9. 27
10. 30
11. 32
12. 32
13. 33
14. 35
15. 37
16. 37
17. 43
18. 46
19. 46
20. 48
21. 49
22. 53
23. 54
24. 61
25. 62
26. 64
27. 64
28. 64
29. 66
30. 66
31. 70
32. 70
33. 72
34. 78
35. 79
36. 79
37. Wait, no, the problem says 36 ages. Wait, maybe I miscounted. Let's check the original data:

Looking at the table, the rows are:

First column (ages?): 17, 19, 23, 24, 26, 26, 27, 27, 27, 30

Second column: 32, 32, 33, 35, 37, 37, 43, 46, 46, 48

Third column: 49, 53, 54, 61, 62, 64, 64, 64, 66, 66

Fourth column: 70, 70, 72, 78, 79, 79, 80? Wait, no, the number of data points: let's count the number of entries. Let's see:

First row: 17, 32, 48, 70? Wait, maybe the table is structured as columns. Let's re-express the data correctly. The problem says "36 sorted ages", so let's list them in order:

1. 17
2. 19
3. 23
4. 24
5. 26
6. 26
7. 27
8. 27
9. 27
10. 30
11. 32
12. 32
13. 33
14. 35
15. 37
16. 37
17. 43
18. 46
19. 46
20. 48
21. 49
22. 53
23. 54
24. 61
25. 62
26. 64
27. 64
28. 64
29. 66
30. 66
31. 70
32. 70
33. 72
34. 78
35. 79
36. 79

Wait, no, that's 36? Wait, 1 - 10: 10, 11 - 20: 10 (total 20), 21 - 30: 10 (total 30), 31 - 36: 6? Wait, maybe I made a mistake. Let's count again. Let's list all the numbers:

From the table, the numbers are:

17, 19, 23, 24, 26, 26, 27, 27, 27, 30,

32, 32, 33, 35, 37, 37, 43, 46, 46, 48,

49, 53, 54, 61, 62, 64, 64, 64, 66, 66,

70, 70, 72, 78, 79, 79, 80? Wait, no, the last number in the fourth column? Wait, the original table has:

First column (vertical): 17, 19, 23, 24, 26, 26, 27, 27, 27, 30

Second column: 32, 32, 33, 35, 37, 37, 43, 46, 46, 48

Third column: 49, 53, 54, 61, 62, 64, 64, 64, 66, 66

Fourth column: 70, 70, 72, 78, 79, 79, 80? Wait, no, the number of rows: let's see, the first column has 10 numbers (17 to 30), second column 10 (32 to 48), third column 10 (49 to 66), fourth column 6? No, that can't be. Wait, the problem says 36 ages, so 36 data points. Let's count the number of entries:

Looking at the table, the numbers are:

17, 32, 48, 70,

19, 32, 53, 72,

23, 33, 54, 78,

24, 35, 61, 79,

26, 37, 62, 79,

26, 37, 64, 80,

27, 43, 64, (missing?),

27, 46, 64, (missing?),

27, 46, 66, (missing?),

30, 48, 66, (missing?),

Wait, maybe the table is 9 rows and 4 columns? 9*4=36. Let's check:

Row 1: 17, 32, 48, 70

Row 2: 19, 32, 53, 72

Row 3: 23, 33, 54, 78

Row 4: 24, 35, 61, 79

Row 5: 26, 37, 62, 79

Row 6: 26, 37, 64, 80

Row 7: 27, 43, 64, (let's see, maybe 64? No, 64 is already there. Wait, maybe I misread. Let's list all 36 numbers:

1. 17

2. 19

3. 23

4. 24

5. 26

6. 26

7. 27

8. 27

9. 27

10. 30

11. 32

12. 32

13. 33

14. 35

15. 37

16. 37

17. 43

18. 46

19. 46

20. 48

21. 49

22. 53

23. 54

24. 61

25. 62

26. 64

27. 64

28. 64

29. 66

30. 66

31. 70

32. 70

33. 72

34. 78

35. 79

36. 79

Wait, no, that's 36? Wait, 1 - 10: 10, 11 - 20: 10 (total 20), 21 - 30: 10 (total 30), 31 - 36: 6? No, 31:70, 32:70, 33:72, 34:78, 35:79, 36:79. Yes, that's 36. Now, the position \( i = 28.8 \), so we round up to 29. The 29th value in the sorted list (from 1 to 36) is the 29th number. Let's count:

1:17, 2:19, 3:23, 4:24, 5:26, 6:26, 7:27, 8:27, 9:27, 10:30,

11:32, 12:32, 13:33, 14:35, 15:37, 16:37, 17:43, 18:46, 19:46, 20:48,

21:49, 22:53, 23:54, 24:61, 25:62, 26:64, 27:64, 28:64, 29:66, 30:66,

31:70, 32:70, 33:72, 34:78, 35:79, 36:79.

Yes, the 29th value is 66. Wait, no, wait: 28th is 64 (26:64, 27:64, 28:64), then 29th is 66 (30th is 66, 31st 70, etc.). Wait, no, let's count again:

1. 17

2. 19

3. 23

4. 24

5. 26

6. 26

7. 27

8. 27

9. 27

10. 30

11. 32

12. 32

13. 33

14. 35

15. 37

16. 37

17. 43

18. 46

19. 46

20. 48

21. 49

22. 53

23. 54

24. 61

25. 62

26. 64

27. 64

28. 64

29. 66

30. 66

31. 70

32. 70

33. 72

34. 78

35. 79

36. 79

Yes, the 29th term is 66. Wait, but let's confirm the formula. The formula for percentile when \( i \) is not an integer is to round up to the next integer and take that value. So \( i = 28.8 \), round up to 29, so the 29th value is 66. Wait, but let's check the data again. Wait, maybe I made a mistake in the data list. Let's check the original table again. The table has:

First column (vertical): 17, 19, 23, 24, 26, 26, 27, 27, 27, 30

Second column: 32, 32, 33, 35, 37, 37, 43, 46, 46, 48

Third column: 49, 53, 54, 61, 62, 64, 64, 64, 66, 66

Fourth column: 70, 70, 72, 78, 79, 79, 80? Wait, no, the fourth column has 7 numbers? No, 10 rows? Wait, maybe the table is 9 rows and 4 columns, but 9*4=36? No, 9*4=36? 9 rows, 4 columns: 9*4=36. Let's list them row-wise:

Row 1: 17, 32, 48, 70

Row 2: 19, 32, 53, 72

Row 3: 23, 33, 54, 78

Row 4: 24, 35, 61, 79

Row 5: 26, 37, 62, 79

Row 6: 26, 37, 64, 80

Row 7: 27, 43, 64, (let's see, maybe 64? No, 64 is in row 6. Wait, row 7: 27, 43, 64, 64?

Row 8: 27, 46, 64, 66

Row 9: 27, 46, 66, 70

Row 10: 30, 48, 70, 72? No, this is getting confusing. Alternatively, maybe the data is already sorted, and we can use the formula \( i = \frac{80}{100} \times 36 = 28.8 \). Since \( i \) is not an integer, we take the value at position \( \lceil i ceil = 29 \). Now, let's count the sorted data correctly. Let's list all 36 numbers in order:

1. 17

2. 19

3. 23

4. 24

5. 26

6. 26

7. 27

8. 27

9. 27

10. 30

11. 32

12. 32

13. 33

14. 35

15. 37

16. 37

17. 43

18. 46

19. 46

20. 48

21. 49

22. 53

23. 54

24. 61

25. 62

26. 64

27. 64

28. 64

29. 66

30. 66

31. 70

32. 70

33. 72

34. 78

35. 79

36. 79

Yes, so the 29th number is 66. Wait, but let's check with another approach. The 80th percentile means 80% of the data is below it. 80% of 36 is 28.8, so we need the value that is greater than 28.8 of the data points. So we take the 29th value, which is 66.

Answer:

\( P_{80} = 66 \)