Question

construct the cumulative frequency distribution for the given data.
age (years) of best actress when award was won | frequency
20 - 29 | 29
30 - 39 | 33
40 - 49 | 14
50 - 59 | 2
60 - 69 | 7
70 - 79 | 1
80 - 89 | 1

age (years) of best actress when award was won | cumulative frequency
less than 30 | 62
less than 40 | 72
less than 50 | 78
less than 60 | 85
less than 70 | 88
less than 80 | 87
less than 90 | 87

Explanation:

To construct the cumulative frequency distribution, we start with the first class and add frequencies successively.

Step 1: Less than 30

The frequency for 20 - 29 is 29. So cumulative frequency for less than 30 is 29.

Step 2: Less than 40

We add the frequency of 30 - 39 (33) to the previous cumulative frequency (29). So \(29 + 33=62\).

Step 3: Less than 50

We add the frequency of 40 - 49 (14) to the previous cumulative frequency (62). So \(62+ 14 = 76\)? Wait, but the given cumulative frequency for less than 50 is 78. Wait, maybe the frequencies are: 20 - 29:29, 30 - 39:33, 40 - 49:14, 50 - 59:2, 60 - 59:7, 70 - 79:1, 80 - 89:1.

Wait, let's recalculate properly.

Cumulative frequency for less than 30: frequency of 20 - 29 = 29.

Cumulative frequency for less than 40: cumulative frequency less than 30 + frequency of 30 - 39 = \(29+33 = 62\) (matches the given 62).

Cumulative frequency for less than 50: cumulative frequency less than 40 + frequency of 40 - 49 = \(62 + 14=76\)? But the given is 78. Wait, maybe the frequency for 40 - 49 is 16? Wait, no, let's check the given data again.

Wait the table has:

Age (years) of Best Actress when award was won: 20 - 29, 30 - 39, 40 - 49, 50 - 59, 60 - 69, 70 - 79, 80 - 89

Frequency: 29, 33, 14, 2, 7, 1, 1

Now, cumulative frequency less than 30: 29

Less than 40: 29 + 33 = 62

Less than 50: 62+14 = 76? But the given cumulative frequency for less than 50 is 78. Wait, maybe there is a typo, or maybe I misread the frequency. Wait, maybe the frequency for 40 - 49 is 16? Let's recalculate with the given cumulative frequencies.

Given cumulative frequency for less than 30: 62? No, the lower part of the image shows:

Less than 30: 62

Less than 40:72

Less than 50:78

Less than 60:85

Less than 70:88

Less than 80:87

Less than 90:87

Wait, maybe the frequencies are different. Let's use the given cumulative frequencies to find the frequencies.

Let's denote the frequencies as \(f_1, f_2, f_3, f_4, f_5, f_6, f_7\) for age groups 20 - 29, 30 - 39, 40 - 49, 50 - 59, 60 - 69, 70 - 79, 80 - 89 respectively.

Cumulative frequency less than 30: \(F_1=f_1 = 62\)? No, that can't be. Wait, maybe the first age group is less than 30, with frequency 29, then less than 40 is 29 + 33 = 62, less than 50 is 62+14 = 76, but the given is 78. So maybe the frequency for 40 - 49 is 16. Then 62 + 16 = 78. Then less than 60: 78 + 2 = 80? But given is 85. So 78 + 7 = 85. Ah, so 50 - 59 frequency is 7? Wait, the original frequency table has 50 - 59:2, 60 - 69:7. So cumulative frequency less than 60: cumulative less than 50 + frequency of 50 - 59 = 78+2 = 80? But given is 85. So 78 + 7 = 85. So maybe the frequency for 50 - 59 is 7 and 60 - 69 is 2? No, the original frequency table (top part) has 50 - 59:2, 60 - 69:7.

This is confusing. Let's proceed with the correct method of cumulative frequency:

Cumulative frequency for a class "less than \(x\)" is the sum of frequencies of all classes with upper limit less than \(x\).

So:

- Less than 30: frequency of 20 - 29 = 29

- Less than 40: frequency of 20 - 29 + frequency of 30 - 39 = \(29 + 33=62\)

- Less than 50: frequency of 20 - 29 + 30 - 39 + 40 - 49 = \(29+33 + 14=76\)

- Less than 60: \(29+33 + 14+2 = 78\)

- Less than 70: \(29+33 + 14+2 + 7=85\)

- Less than 80: \(29+33 + 14+2 + 7+1 = 86\)

- Less than 90: \(29+33 + 14+2 + 7+1 + 1=87\)

Ah, now we can see the correct cumulative frequencies:

- Less than 30: 29

- Less than 40: 29 + 33 = 62

- Less than 50: 62 + 14 = 76? But the given in the lower table is 78. Wait, maybe the frequency for 40 - 49 is 16. Then 62 + 16 = 78. Then less than 60: 78 + 2 = 80, but given is 85. So 78 + 7 = 85. So maybe the frequency for 50 - 59 is 7 and 60 - 69 is 2. Let's adjust the frequencies:

Suppose:

20 - 29:29

30 - 39:33

40 - 49:16 (so that 29+33+16=78)

50 - 59:7 (78+7=85)

60 - 69:3 (85+3=88)

70 - 79: -1? No, that's impossible.

Wait, the lower part of the image shows cumulative frequencies:

Less than 30:62

Less than 40:72

Less than 50:78

Less than 60:85

Less than 70:88

Less than 80:87

Less than 90:87

Let's find the frequencies from cumulative frequencies.

Let \(F_1\) = cumulative less than 30 = 62

\(F_2\) = cumulative less than 40 = 72, so frequency of 30 - 39 is \(F_2 - F_1=72 - 62 = 10\)

\(F_3\) = cumulative less than 50 = 78, so frequency of 40 - 49 is \(F_3 - F_2=78 - 72 = 6\)

\(F_4\) = cumulative less than 60 = 85, so frequency of 50 - 59 is \(F_4 - F_3=85 - 78 = 7\)

\(F_5\) = cumulative less than 70 = 88, so frequency of 60 - 69 is \(F_5 - F_4=88 - 85 = 3\)

\(F_6\) = cumulative less than 80 = 87, so frequency of 70 - 79 is \(F_6 - F_5=87 - 88=-1\) (impossible)

\(F_7\) = cumulative less than 90 = 87, so frequency of 80 - 89 is \(F_7 - F_6=87 - 87 = 0\)

This is inconsistent, so there must be a misprint in the given data. But assuming the initial frequency table is:

Age: 20 - 29, 30 - 39, 40 - 49, 50 - 59, 60 - 69, 70 - 79, 80 - 89

Frequency:29, 33, 14, 2, 7, 1, 1

Then cumulative frequencies are:

Less than 30:29

Less than 40:29 + 33 = 62

Less than 50:62 + 14 = 76

Less than 60:76 + 2 = 78

Less than 70:78 + 7 = 85

Less than 80:85 + 1 = 86

Less than 90:86 + 1 = 87

Which matches the last cumulative frequency (87 for less than 90). The given cumulative frequencies in the image for less than 50 is 78 (maybe a typo, should be 76), less than 60 is 85 (matches 78 + 7=85), less than 70 is 88 (should be 85 + 1=86? No, 85 + 3=88, so maybe frequency of 60 - 69 is 3).

But the correct way to construct cumulative frequency is to add the frequencies successively. So the cumulative frequency distribution is:

| Age (less than) | Cumulative Frequency |
|-----------------|----------------------|
| Less than 30 | 29 |
| Less than 40 | 29 + 33 = 62 |
| Less than 50 | 62 + 14 = 76 |
| Less than 60 | 76 + 2 = 78 |
| Less than 70 | 78 + 7 = 85 |
| Less than 80 | 85 + 1 = 86 |
| Less than 90 | 86 + 1 = 87 |

Answer:

The cumulative frequency distribution is constructed by adding the frequencies of each age group successively. The cumulative frequencies are: Less than 30: 29, Less than 40: 62, Less than 50: 76, Less than 60: 78, Less than 70: 85, Less than 80: 86, Less than 90: 87 (assuming the given frequency data: 20 - 29:29, 30 - 39:33, 40 - 49:14, 50 - 59:2, 60 - 69:7, 70 - 79:1, 80 - 89:1).