Question

as of 2011, the following are the ages at which a countrys presidents were inaugurated.
53, 60, 52, 67, 60, 55, 51, 47, 48, 49, 62, 50, 63, 69, 68, 62, 69, 67,
44, 46, 64, 48, 47, 59, 55, 60, 68, 60, 61, 61, 57, 56, 49, 46, 48,
61, 64, 44, 51, 58, 59, 55, 42
construct a grouped frequency distribution for the data. use 41 - 45 for the first class and use the same width for each subsequent class.
type the frequency value for each class.
age
41 - 45
46 - 50
51 - 55
frequency
3
11
□

Explanation:

Step1: Identify the class width

The first class is 41 - 45, so the class width is \( 45 - 41 + 1 = 5 \)? Wait, no, actually, for grouped frequency, the class width is the difference between the upper and lower bounds. Wait, the problem says "use the same width for each subsequent class". The first class is 41 - 45, so the width is \( 45 - 41 = 4 \)? Wait, no, 41 to 45 is 5 numbers (41,42,43,44,45), so the class width is 5? Wait, maybe the classes are 41 - 45, 46 - 50, 51 - 55, etc., so each class has a width of 5 (since 46 - 41 = 5, 51 - 46 = 5, etc.). So we need to count the number of data points in the 51 - 55 class.

First, let's list all the data points:

53, 60, 52, 62, 60, 55, 51, 47, 48, 49, 62, 50, 63, 69, 68, 62, 69, 67, 44, 46, 64, 48, 47, 59, 55, 60, 68, 60, 61, 61, 57, 47, 56, 49, 46, 48, 61, 64, 44, 51, 58, 59, 55, 42.

Now, let's list the data points in the 51 - 55 class. The lower bound is 51, upper bound is 55 (inclusive? Or exclusive? Since the previous class is 46 - 50, so 51 - 55 would include 51,52,53,54,55.

Let's find all data points between 51 and 55 (inclusive):

51 (appears: let's check each number:

53: yes

52: yes

51: yes (from "51" in the list)

55: yes (multiple times)

51: another "51"

53: yes

52: yes

55: yes

51: yes

Wait, let's list all:

Looking at the data:

53, 52, 51, 55, 51, 53, 52, 55, 51, 55, 55, 51? Wait, no, let's go through each data point:

1. 53: 51 - 55: yes
2. 60: no
3. 52: yes
4. 62: no
5. 60: no
6. 55: yes
7. 51: yes
8. 47: no
9. 48: no
10. 49: no
11. 62: no
12. 50: no
13. 63: no
14. 69: no
15. 68: no
16. 62: no
17. 69: no
18. 67: no
19. 44: no
20. 46: no
21. 64: no
22. 48: no
23. 47: no
24. 59: no
25. 55: yes
26. 60: no
27. 68: no
28. 60: no
29. 61: no
30. 61: no
31. 57: no
32. 47: no
33. 56: no
34. 49: no
35. 46: no
36. 48: no
37. 61: no
38. 64: no
39. 44: no
40. 51: yes
41. 58: no
42. 59: no
43. 55: yes
44. 42: no

Now let's count these:

53 (1), 52 (1), 55 (26), 51 (7), 55 (25), 51 (40), 55 (43), 53 (1), 52 (3), 55 (6), 51 (7), 55 (25), 51 (40), 55 (43). Wait, no, let's list each valid one:

- 51: positions 7, 40 → 2 times? Wait, no, let's list all the data points again carefully:

Wait, the data list:

1. 53

2. 60

3. 52

4. 62

5. 60

6. 55

7. 51

8. 47

9. 48

10. 49

11. 62

12. 50

13. 63

14. 69

15. 68

16. 62

17. 69

18. 67

19. 44

20. 46

21. 64

22. 48

23. 47

24. 59

25. 55

26. 60

27. 68

28. 60

29. 61

30. 61

31. 57

32. 47

33. 56

34. 49

35. 46

36. 48

37. 61

38. 64

39. 44

40. 51

41. 58

42. 59

43. 55

44. 42

Now, let's extract numbers between 51 and 55 (inclusive):

- 51: position 7, 40 → 2 times? Wait, position 7: 51, position 40: 51 → 2?

- 52: position 3 → 1

- 53: position 1 → 1

- 55: position 6, 25, 43 → 3 times? Wait, position 6: 55, position 25: 55, position 43: 55 → 3?

Wait, no, let's check each number:

51: 7, 40 → 2

52: 3 → 1

53: 1 → 1

55: 6, 25, 43 → 3

Wait, but that's 2 + 1 + 1 + 3 = 7? Wait, maybe I missed some.

Wait, let's list all numbers in 51 - 55:

51, 52, 53, 54, 55.

Looking at the data:

51: 7, 40 → 2

52: 3 → 1

53: 1 → 1

55: 6, 25, 43 → 3

Wait, is there a 54? No.

Wait, maybe the class is 51 - 55, so numbers from 51 to 55, inclusive. Let's count again:

Data points:

53 (1), 52 (3), 55 (6), 51 (7), 55 (25), 51 (40), 55 (43), 53 (1)? Wait, no, the data list is:

1. 53

3. 52

6. 55

7. 51

40. 51

25. 55

43. 55

Wait, that's 53 (1), 52 (1), 55 (3), 51 (2). Wait, 1 + 1 + 3 + 2 = 7? Wait, maybe I made a mistake. Let's list all data points and mark those in 51 - 55:

1. 53 – yes

2. 60 – no

3. 52 – yes

4. 62 – no

5. 60 – no

6. 55 – yes

7. 51 – yes

8. 47 – no

9. 48 – no

10. 49 – no

11. 62 – no

12. 50 – no

13. 63 – no

14. 69 – no

15. 68 – no

16. 62 – no

17. 69 – no

18. 67 – no

19. 44 – no

20. 46 – no

21. 64 – no

22. 48 – no

23. 47 – no

24. 59 – no

25. 55 – yes

26. 60 – no

27. 68 – no

28. 60 – no

29. 61 – no

30. 61 – no

31. 57 – no

32. 47 – no

33. 56 – no

34. 49 – no

35. 46 – no

36. 48 – no

37. 61 – no

38. 64 – no

39. 44 – no

40. 51 – yes

41. 58 – no

42. 59 – no

43. 55 – yes

44. 42 – no

Now count the "yes" ones:

1. 53

3. 52

6. 55

7. 51

25. 55

40. 51

43. 55

Wait, that's 7 data points? Wait, 1 (53) + 1 (52) + 3 (55s) + 2 (51s) = 7. So the frequency for 51 - 55 is 7? Wait, but let's check the total number of data points. Let's count the number of data points:

Let's count the number of numbers:

First row: 53, 60, 52, 62, 60, 55, 51, 47, 48, 49, 62, 50, 63, 69, 68, 62, 69, 67 → 18

Second row: 44, 46, 64, 48, 47, 59, 55, 60, 68, 60, 61, 61, 57, 47, 56, 49, 46, 48 → 18

Third row: 61, 64, 44, 51, 58, 59, 55, 42 → 8

Total: 18 + 18 + 8 = 44 data points.

Now, the frequency for 41 - 45: let's count. The 41 - 45 class includes 41,42,43,44,45.

Data points in 41 - 45:

44 (19), 44 (39), 42 (44), 44 (19), 44 (39), 42 (44). Wait, data points:

19. 44 – yes

39. 44 – yes

44. 42 – yes

Any others? 41? No. 43? No. 45? No. So 44 (2), 42 (1) → total 3, which matches the given frequency of 3 for 41 - 45. Good.

Frequency for 46 - 50: let's count. 46 - 50 includes 46,47,48,49,50.

Data points in 46 - 50:

47 (8), 48 (9), 49 (10), 48 (22), 47 (23), 49 (34), 46 (20), 48 (36), 47 (32), 46 (35), 48 (36? Wait, no, let's list all:

46 (20), 46 (35) → 2

47 (8), 47 (23), 47 (32) → 3

48 (9), 48 (22), 48 (36) → 3

49 (10), 49 (34) → 2

50 (12) → 1

Total: 2 + 3 + 3 + 2 + 1 = 11, which matches the given frequency of 11 for 46 - 50. Good.

Now, total frequency so far: 3 (41 - 45) + 11 (46 - 50) = 14. So the remaining frequency for 51 - 55 and above should be 44 - 14 = 30? Wait, no, because there are more classes, but the problem only shows 41 - 45 (3), 46 - 50 (11), and 51 - 55 (unknown). Wait, maybe the total number of data points is 44, so 3 + 11 + x = 44? No, because there are more classes (like 56 - 60, 61 - 65, etc.), but the problem only asks for 51 - 55. Wait, no, the table has Age: 41 - 45, 46 - 50, 51 - 55, so we need to find the frequency for 51 - 55.

We know total data points: 44.

Frequency for 41 - 45: 3

Frequency for 46 - 50: 11

So frequency for 51 - 55: let's count again carefully.

Data points in 51 - 55:

51 (7), 51 (40) → 2

52 (3) → 1

53 (1) → 1

55 (6), 55 (25), 55 (43) → 3

Wait, 2 + 1 + 1 + 3 = 7? Wait, but let's list all:

51 (7), 51 (40) → 2

52 (3) → 1

53 (1) → 1

55 (6), 55 (25), 55 (43) → 3

Total: 2 + 1 + 1 + 3 = 7. Let's check with total: 3 (41 - 45) + 11 (46 - 50) + 7 (51 - 55) = 21. Then the remaining data points (44 - 21 = 23) would be in 56 - 60, 61 - 65, etc. But maybe my count is wrong.

Wait, let's list all data points in 51 - 55:

51, 51, 52, 53, 55, 55, 55. Wait, that's 7 numbers? Wait, 51 (2), 52 (1), 53 (1), 55 (3) → total 7. Yes, that's 7. So the frequency for 51 - 55 is 7.

Step2: Confirm the count

After carefully identifying and counting the data points within the 51 - 55 age range, we find that there are 7 such data points.

Answer:

7