Question

a random sample of 30 male college students was selected, and their heights (in inches) are given below.
74 69 74 73 67 69
70 73 73 67 67 70
68 73 73 70 69 partially visible
70 67 72 67 74 partially visible
67 72 66 71 70 72
(a) complete the frequency distribution for the data. make sure to enter your frequency as decimals, rounded to the nearest hundredth.

height	frequency	relative frequency
67	6 ✔️	0.20 ✔️
68	1 ✔️	0.3 ❌
69	3 ✔️	0.11 ✔️
70	5 ✔️	0.17 ✔️
71	2 ✔️	0.07 ✔️
72	3 ❌	0.10 ❌
73	5 ✔️	0.17 ✔️
74	2 ❌	0.7 ❌

Explanation:

Step1: Recall Relative Frequency Formula

Relative Frequency = $\frac{\text{Frequency}}{\text{Total Number of Observations}}$. Total number of students is 30.

Step2: Correct Relative Frequency for 68

Frequency of 68 is 1. So Relative Frequency = $\frac{1}{30} \approx 0.03$ (rounded to nearest hundredth).

Step3: Count Frequency for 72

Looking at the data: 72 appears at positions (row 4, column 3), (row 5, column 2), (row 5, column 6). Wait, let's list all 72s: 72, 72, 72? Wait original data:

Row 1: 74,69,74,73,67,69

Row 2:70,73,73,67,67,70

Row 3:68,73,73,70,69, (missing? Wait total 30. Let's recount all heights:

66:1

67: Let's count: 67 (row1,col5), 67 (row2,col4), 67 (row2,col5), 67 (row4,col2), 67 (row4,col4), 67 (row5,col1) → 6. Correct.

68:1 (row3,col1)

69: row1,col2; row1,col6; row3,col5 → 3. Correct.

70: row2,col1; row2,col6; row3,col4; row4,col1; row5,col5 → 5. Correct.

71: row5,col4 → 1? Wait no, original data row5: 67,72,66,71,70,72. Wait row4:70,67,72,67,74, (missing? Wait total 30. Let's list all 30 numbers:

Row1 (6): 74,69,74,73,67,69

Row2 (6):70,73,73,67,67,70

Row3 (6):68,73,73,70,69, (wait, maybe a typo, but let's check row4:70,67,72,67,74, (maybe 72? No, row4: 70,67,72,67,74, (let's count total so far: 6+6+6+6+6=30. Row4: 70,67,72,67,74, x. Row5:67,72,66,71,70,72.

Wait let's list all:

66:1 (row5,col3)

67: row1,col5; row2,col4; row2,col5; row4,col2; row4,col4; row5,col1 → 6.

68: row3,col1 →1.

69: row1,col2; row1,col6; row3,col5 →3.

70: row2,col1; row2,col6; row3,col4; row4,col1; row5,col5 →5.

71: row5,col4 →1? Wait no, row5: 67,72,66,71,70,72 → 71 is 1? But earlier it was marked 2. Wait maybe I miscounted. Wait original table:

Row 4: 70, 67, 72, 67, 74, (maybe 72? No, row4 has 6 entries: 70,67,72,67,74, and one more? Wait the first table:

First table (heights):

Row 1: 74,69,74,73,67,69 (6)

Row 2:70,73,73,67,67,70 (6)

Row 3:68,73,73,70,69, (let's say 72? No, the third row in the first table: 68,73,73,70,69, (maybe 72? Then row4:70,67,72,67,74, (72? No, row4:70,67,72,67,74, (72? Then row5:67,72,66,71,70,72 (6). Wait total:

Row1:6, row2:6, row3:6, row4:6, row5:6 → 30.

Now count 72:

Row3: maybe 72? No, original row3: 68,73,73,70,69, (let's check the second table's 72 frequency. Wait the error is in 72,74.

Wait 72: row3 (if there's a 72), row4 (72), row5 (72,72). Wait no, let's look at the data again:

From the first table (heights):

- 72 appears at: row4, col3; row5, col2; row5, col6. Wait row4: 70,67,72,67,74, (another 72? No, row4 has 6 entries: 70,67,72,67,74, and one more? Maybe row3, col6 is 72? Let's count all 72s:

Looking at the second table, 72's frequency was marked 3 (wrong). Let's count:

Row4: 72 (1)

Row5:72 (1), 72 (1) → that's 3? No, row5: 67,72,66,71,70,72 → two 72s. Row4:72 → one. So total 3? Wait no, row3: 68,73,73,70,69, (maybe 72? Then row3 has a 72. So row3: 72 (1), row4:72 (1), row5:72 (2). Total 4? Wait this is confusing. Wait the total number of observations is 30. Let's sum all frequencies:

66:1, 67:6, 68:1, 69:3, 70:5, 71:?, 72:?, 73:5, 74:?.

Sum so far:1+6+1+3+5+5=21. So 71 +72 +74=9.

Looking at the data:

71: row5, col4 →1.

74: row1, col1; row1, col3; row4, col5 →3.

So 71:1, 74:3. Then 72: 9 -1 -3=5? No, 21 +1 (71) +x (72) +3 (74)=30 → x=5. Wait that can't be. Wait maybe I made a mistake. Let's list all 30 numbers:

1. 74

2. 69

3. 74

4. 73

5. 67

6. 69

7. 70

8. 73

9. 73

10. 67

11. 67

12. 70

13. 68

14. 73

15. 73

16. 70

17. 69

18. (missing? Wait no, the first table has 5 rows? Wait the first table:

Row 1: 6 cells

Row 2:6 cells

Row 3:6 cells

Row 4:6 cells

Row 5:6 cells → 5*6=30.

Row 4: 70,67,72,67,74, (let's say 72? No, row4: 70,67,72,67,74, 72 → that's 6.

Row 5:67,72,66,71,70,72 → 6.

Now list all 30:

1.74

2.69

3.74

4.73

5.67

6.69

7.70

8.73

9.73

10.67

11.67

12.70

13.68

14.73

15.73

16.70

17.69

18.70 (row4, col1)

19.67 (row4, col2)

20.72 (row4, col3)

21.67 (row4, col4)

22.74 (row4, col5)

23.72 (row4, col6)

24.67 (row5, col1)

25.72 (row5, col2)

26.66 (row5, col3)

27.71 (row5, col4)

28.70 (row5, col5)

29.72 (row5, col6)

Now count each height:

66:1 (26)

67:5,10,11,19,21,24 → 6 (correct)

68:13 →1 (correct)

69:2,6,17 →3 (correct)

70:7,12,16,18,28 →5 (correct)

71:27 →1 (so frequency 1, not 2. Wait original table had 71 frequency as 2, which is wrong. But let's focus on 68,72,74.

68: frequency 1, relative frequency = 1/30 ≈0.03 (rounded to nearest hundredth).

72: count the 72s:20,23,25,29 → wait 20 (row4,col3),23 (row4,col6),25 (row5,col2),29 (row5,col6) → that's 4? Wait no, 20,23,25,29: four? Wait row4,col3:72, row4,col6:72, row5,col2:72, row5,col6:72 → four. Wait but earlier sum:1+6+1+3+5+1 (71) +4 (72) +3 (74)=24? No, 1+6=7, +1=8, +3=11, +5=16, +1=17, +4=21, +3=24. Missing 6. Oh, 73: let's count 73s:4,8,9,14,15 →5 (correct). So 16 (70) +5 (73)=21, +3 (74)=24, +1 (71)=25, +4 (72)=29, +1 (66)=30, +6 (67)=36? No, I'm messing up.

Wait let's use the correct method: for each height, count how many times it appears.

Height 66: appears once (row5, col3) → frequency 1.

Height 67: let's list all 67s: row1,col5; row2,col4; row2,col5; row4,col2; row4,col4; row5,col1 → 6 times → frequency 6.

Height 68: row3,col1 →1 time → frequency 1.

Height 69: row1,col2; row1,col6; row3,col5 →3 times → frequency 3.

Height 70: row2,col1; row2,col6; row3,col4; row4,col1; row5,col5 →5 times → frequency 5.

Height 71: row5,col4 →1 time → frequency 1 (original had 2, wrong).

Height 72: row4,col3; row4,col6; row5,col2; row5,col6 →4 times? Wait row4: 70,67,72,67,74,72 → two 72s. Row5:67,72,66,71,70,72 → two 72s. Total 4 → frequency 4.

Height 73: row1,col4; row2,col2; row2,col3; row3,col2; row3,col3 →5 times → frequency 5 (correct).

Height 74: row1,col1; row1,col3; row4,col5 →3 times → frequency 3 (original had 2, wrong).

Now, relative frequency is frequency /30.

For 68: 1/30 ≈0.03 (rounded to nearest hundredth).

For 72: 4/30 ≈0.13 (rounded to nearest hundredth).

For 74: 3/30 =0.10 (rounded to nearest hundredth).

For 71:1/30≈0.03 (rounded to nearest hundredth).

But the problem's table has some errors. Let's correct the ones marked:

- 68: relative frequency was 0.3 (wrong), correct is 1/30≈0.03.

- 72: frequency was 3 (wrong), correct frequency is 4; relative frequency 4/30≈0.13.

- 74: frequency was 2 (wrong), correct frequency is 3; relative frequency 3/30=0.10.

- 71: frequency was 2 (wrong), correct frequency is 1; relative frequency 1/30≈0.03.

But the question is to complete the frequency distribution, so let's focus on the marked errors:

68:

Frequency =1, Relative Frequency = 1/30 ≈0.03.

72:

Count 72s: row4 (72,72), row5 (72,72) → 4? Wait no, row4: 70,67,72,67,74,72 (two 72s). Row5:67,72,66,71,70,72 (two 72s). Total 4. So frequency=4, Relative Frequency=4/30≈0.13.

74:

74s: row1 (74,74), row4 (74) → three 74s. So frequency=3, Relative Frequency=3/30=0.10.

71:

71: row5,col4 →1. So frequency=1, Relative Frequency=1/30≈0.03.

But the problem's table may have a typo in the data, but based on the given data (assuming row3,col6 is 72, row4,col6 is 72, etc.), let's correct the relative frequency for 68 first.

Answer:

For height 68: Relative Frequency = $\frac{1}{30} \approx 0.03$ (rounded to nearest hundredth).

For height 72: Frequency = 4, Relative Frequency = $\frac{4}{30} \approx 0.13$.

For height 74: Frequency = 3, Relative Frequency = $\frac{3}{30} = 0.10$.

For height 71: Frequency = 1, Relative Frequency = $\frac{1}{30} \approx 0.03$.

(But focusing on the marked error for 68: correct relative frequency is 0.03)