Question

modeling real life the heart rates (in beats per minute) for a random sample of 36 blue whales are shown in the table. blue whale heart rates 11 10 12 12 8 10 13 11 13 9 11 8 12 6 8 13 5 8 9 8 12 6 9 7 6 8 7 9 9 5 12 9 11 7 8 9 estimate the population mean \\( \mu \\). round your answer to the nearest hundredth. the population mean \\( \mu \\) is about \\( \square \\).

Explanation:

Step1: Sum all heart rates

First, we list all the heart rates and sum them up. Let's count each value:

5 appears 2 times: \(5\times2 = 10\)

6 appears 3 times: \(6\times3 = 18\)

7 appears 3 times: \(7\times3 = 21\)

8 appears 7 times: \(8\times7 = 56\)

9 appears 7 times: \(9\times7 = 63\)

10 appears 2 times: \(10\times2 = 20\)

11 appears 3 times: \(11\times3 = 33\)

12 appears 4 times: \(12\times4 = 48\)

13 appears 3 times: \(13\times3 = 39\)

Now sum all these products: \(10 + 18 + 21 + 56 + 63 + 20 + 33 + 48 + 39\)

Calculate step by step:

\(10+18 = 28\); \(28+21 = 49\); \(49+56 = 105\); \(105+63 = 168\); \(168+20 = 188\); \(188+33 = 221\); \(221+48 = 269\); \(269+39 = 308\)

Wait, wait, maybe I miscounted the number of each value. Let's count again by looking at the table:

First row: 11, 10, 12, 12, 8, 10 (values: 11,10,12,12,8,10)

Second row:13,11,13,9,11,8 (13,11,13,9,11,8)

Third row:12,6,8,13,5,8 (12,6,8,13,5,8)

Fourth row:9,8,12,6,9,7 (9,8,12,6,9,7)

Fifth row:6,8,7,9,9,5 (6,8,7,9,9,5)

Sixth row:12,9,11,7,8,9 (12,9,11,7,8,9)

Now let's list all 36 values:

11,10,12,12,8,10,

13,11,13,9,11,8,

12,6,8,13,5,8,

9,8,12,6,9,7,

6,8,7,9,9,5,

12,9,11,7,8,9

Now count each number:

5: positions (3rd row, 5th column), (5th row, 6th column) → 2 times.

6: (3rd row, 2nd column), (4th row, 4th column), (5th row, 1st column) → 3 times? Wait 3rd row: 12,6,8,13,5,8 → 6 (1), 4th row:9,8,12,6,9,7 → 6 (2), 5th row:6,8,7,9,9,5 → 6 (3). So 3 times.

7: (4th row, 6th column), (5th row, 3rd column), (6th row, 4th column) → 3 times? 4th row:9,8,12,6,9,7 →7 (1), 5th row:6,8,7,9,9,5 →7 (2), 6th row:12,9,11,7,8,9 →7 (3). So 3 times.

8: (1st row,5th), (2nd row,6th), (3rd row,3rd), (3rd row,6th), (4th row,2nd), (5th row,2nd), (6th row,5th) → let's count:

1st row:8 (1)

2nd row:8 (2)

3rd row:8 (3), 8 (4)

4th row:8 (5)

5th row:8 (6)

6th row:8 (7) → 7 times. Correct.

9: (2nd row,4th), (4th row,1st), (4th row,5th), (5th row,4th), (5th row,5th), (6th row,2nd), (6th row,6th) →

2nd row:9 (1)

4th row:9 (2), 9 (3)

5th row:9 (4), 9 (5)

6th row:9 (6), 9 (7) → 7 times. Correct.

10: (1st row,2nd), (1st row,6th) → 2 times.

11: (1st row,1st), (2nd row,2nd), (2nd row,5th), (6th row,3rd) →

1st row:11 (1)

2nd row:11 (2), 11 (3)

6th row:11 (4) → Wait, 1st row:11, 2nd row:11,11, 6th row:11 → that's 4 times? Wait 1st row:11 (1), 2nd row:13,11,13,9,11,8 → 11 (2), 11 (3), 6th row:12,9,11,7,8,9 → 11 (4). So 4 times? Wait my initial count was wrong. Oh no, I made a mistake earlier. Let's recount 11:

1st row:11 (1)

2nd row:11 (2), 11 (3)

6th row:11 (4) → 4 times.

12: (1st row,3rd), (1st row,4th), (3rd row,1st), (4th row,3rd), (6th row,1st) →

1st row:12 (1), 12 (2)

3rd row:12 (3)

4th row:12 (4)

6th row:12 (5) → Wait 1st row:12,12; 3rd row:12; 4th row:12; 6th row:12 → 5 times? Wait 1st row:12 (1), 12 (2); 3rd row:12 (3); 4th row:12 (4); 6th row:12 (5) → 5 times? Wait 1st row:11,10,12,12,8,10 → two 12s. 3rd row:12,6,8,13,5,8 → one 12. 4th row:9,8,12,6,9,7 → one 12. 6th row:12,9,11,7,8,9 → one 12. So total 2+1+1+1=5? Wait no, 1st row:2, 3rd:1, 4th:1, 6th:1 → 5. But earlier I thought 4. So my mistake.

13: (2nd row,1st), (2nd row,3rd), (3rd row,4th) →

2nd row:13 (1), 13 (2)

3rd row:13 (3) → 3 times. Correct.

Now let's recalculate the sum with correct counts:

5: 2 → \(5\times2 = 10\)

6: 3 → \(6\times3 = 18\)

7: 3 → \(7\times3 = 21\)

8: 7 → \(8\times7 = 56\)

9: 7 → \(9\times7 = 63\)

10: 2 → \(10\times2 = 20\)

11: 4 → \(11\times4 = 44\) (because 1st row:1, 2nd row:2, 6th row:1 → total 4)

12: 5 → \(12\times5 = 60\) (1st row:2, 3rd row:1, 4th row:1, 6th row:1 → 5)

13: 3 → \(13\times3 = 39\)

Now sum all these:

\(10 + 18 = 28\)

\(28 + 21 = 49\)

\(49 + 56 = 105\)

\(105 + 63 = 168\)

\(168 + 20 = 188\)

\(188 + 44 = 232\)

\(232 + 60 = 292\)

\(292 + 39 = 331\)

Wait, but there are 36 values. Let's check the count of each number:

5:2, 6:3, 7:3, 8:7, 9:7, 10:2, 11:4, 12:5, 13:3. Let's sum the counts: 2+3+3+7+7+2+4+5+3 = 36. Correct.

Now sum of all values: 331? Wait let's add the numbers one by one to verify.

List all 36 numbers:

5,5,

6,6,6,

7,7,7,

8,8,8,8,8,8,8,

9,9,9,9,9,9,9,

10,10,

11,11,11,11,

12,12,12,12,12,

13,13,13

Now sum:

5+5=10

+6+6+6=10+18=28

+7+7+7=28+21=49

+8*7=49+56=105

+9*7=105+63=168

+10*2=168+20=188

+11*4=188+44=232

+12*5=232+60=292

+13*3=292+39=331. Correct.

Step2: Calculate the mean

The mean \(\bar{x}\) (which estimates the population mean \(\mu\)) is the sum of all values divided by the number of values (n=36).

So \(\mu \approx \frac{331}{36}\)

Calculate \(\frac{331}{36} \approx 9.1944...\)

Rounded to the nearest hundredth, that's 9.19. Wait, but wait, maybe I made a mistake in counting the number of 11s and 12s. Let's list all the numbers again and count:

First row (6 numbers): 11,10,12,12,8,10 → 11(1),10(1),12(1),12(2),8(1),10(2)

Second row (6):13(1),11(2),13(2),9(1),11(3),8(2)

Third row (6):12(3),6(1),8(3),13(3),5(1),8(4)

Fourth row (6):9(2),8(5),12(4),6(2),9(3),7(1)

Fifth row (6):6(3),8(6),7(2),9(4),9(5),5(2)

Sixth row (6):12(5),9(6),11(4),7(3),8(7),9(7)

Now count 11: 1,2,3,4 → 4 times. Correct.

12:1,2,3,4,5 → 5 times. Correct.

Now sum all numbers:

11+10+12+12+8+10 = 11+10=21, +12=33, +12=45, +8=53, +10=63

13+11+13+9+11+8 =13+11=24, +13=37, +9=46, +11=57, +8=65. Total so far:63+65=128

12+6+8+13+5+8 =12+6=18, +8=26, +13=39, +5=44, +8=52. Total:128+52=180

9+8+12+6+9+7 =9+8=17, +12=29, +6=35, +9=44, +7=51. Total:180+51=231

6+8+7+9+9+5 =6+8=14, +7=21, +9=30, +9=39, +5=44. Total:231+44=275

12+9+11+7+8+9 =12+9=21, +11=32, +7=39, +8=47, +9=56. Total:275+56=331. Yes! So the sum is 331.

Now number of values n=36.

Mean = \( \frac{331}{36} \approx 9.1944\)

Rounded to the nearest hundredth: 9.19 (since the thousandth digit is 4, which is less than 5, we round down the hundredth digit). Wait, 331 divided by 36: 36*9=324, 331-324=7, so 9 + 7/36 ≈9.1944, so to the nearest hundredth, 9.19.

Wait but let me check again. 7 divided by 36: 7÷36≈0.1944, so 9.1944, so hundredth place is 9, thousandth is 4, so round to 9.19.

Answer:

\(9.19\)