Question

b. the principal decides to take 20 different samples of 20 test scores, and calculate the sample mean for each. the sample means are listed below.
77.4, 73.5, 81, 83.2, 79, 77.8, 84.1, 76, 75.8, 82.1, 80.4, 77.1, 78.8, 79.2, 81.2, 75.4, 85.1, 81.2, 77.4, 80.8
what is the mean of the sample means?
(express your answer to the thousandths.)

Explanation:

Step1: Sum all sample means

First, we add up all the given sample means:
\[ \begin{align*} &77.4 + 73.5 + 81 + 83.2 + 79 + 77.8 + 84.1 + 76 + 75.8 + 82.1 + 80.4 + 77.1 + 78.8 + 79.2 + 81.2 + 75.4 + 85.1 + 81.2 + 77.4 + 80.8\\ =& (77.4+73.5)+81 + 83.2 + 79 + 77.8 + 84.1 + 76 + 75.8 + 82.1 + 80.4 + 77.1 + 78.8 + 79.2 + 81.2 + 75.4 + 85.1 + 81.2 + 77.4 + 80.8\\ =& 150.9+81 + 83.2 + 79 + 77.8 + 84.1 + 76 + 75.8 + 82.1 + 80.4 + 77.1 + 78.8 + 79.2 + 81.2 + 75.4 + 85.1 + 81.2 + 77.4 + 80.8\\ =& 231.9+83.2 + 79 + 77.8 + 84.1 + 76 + 75.8 + 82.1 + 80.4 + 77.1 + 78.8 + 79.2 + 81.2 + 75.4 + 85.1 + 81.2 + 77.4 + 80.8\\ =& 315.1+79 + 77.8 + 84.1 + 76 + 75.8 + 82.1 + 80.4 + 77.1 + 78.8 + 79.2 + 81.2 + 75.4 + 85.1 + 81.2 + 77.4 + 80.8\\ =& 394.1+77.8 + 84.1 + 76 + 75.8 + 82.1 + 80.4 + 77.1 + 78.8 + 79.2 + 81.2 + 75.4 + 85.1 + 81.2 + 77.4 + 80.8\\ =& 471.9+84.1 + 76 + 75.8 + 82.1 + 80.4 + 77.1 + 78.8 + 79.2 + 81.2 + 75.4 + 85.1 + 81.2 + 77.4 + 80.8\\ =& 556+76 + 75.8 + 82.1 + 80.4 + 77.1 + 78.8 + 79.2 + 81.2 + 75.4 + 85.1 + 81.2 + 77.4 + 80.8\\ =& 632+75.8 + 82.1 + 80.4 + 77.1 + 78.8 + 79.2 + 81.2 + 75.4 + 85.1 + 81.2 + 77.4 + 80.8\\ =& 707.8+82.1 + 80.4 + 77.1 + 78.8 + 79.2 + 81.2 + 75.4 + 85.1 + 81.2 + 77.4 + 80.8\\ =& 789.9+80.4 + 77.1 + 78.8 + 79.2 + 81.2 + 75.4 + 85.1 + 81.2 + 77.4 + 80.8\\ =& 870.3+77.1 + 78.8 + 79.2 + 81.2 + 75.4 + 85.1 + 81.2 + 77.4 + 80.8\\ =& 947.4+78.8 + 79.2 + 81.2 + 75.4 + 85.1 + 81.2 + 77.4 + 80.8\\ =& 1026.2+79.2 + 81.2 + 75.4 + 85.1 + 81.2 + 77.4 + 80.8\\ =& 1105.4+81.2 + 75.4 + 85.1 + 81.2 + 77.4 + 80.8\\ =& 1186.6+75.4 + 85.1 + 81.2 + 77.4 + 80.8\\ =& 1262+85.1 + 81.2 + 77.4 + 80.8\\ =& 1347.1+81.2 + 77.4 + 80.8\\ =& 1428.3+77.4 + 80.8\\ =& 1505.7+80.8\\ =& 1586.5 \end{align*} \]
Wait, actually, let's do this more carefully. Let's count the number of terms: there are 20 sample means. Let's list them again and sum:

1. 77.4
2. 73.5 (sum: 150.9)
3. 81 (231.9)
4. 83.2 (315.1)
5. 79 (394.1)
6. 77.8 (471.9)
7. 84.1 (556)
8. 76 (632)
9. 75.8 (707.8)
10. 82.1 (789.9)
11. 80.4 (870.3)
12. 77.1 (947.4)
13. 78.8 (1026.2)
14. 79.2 (1105.4)
15. 81.2 (1186.6)
16. 75.4 (1262)
17. 85.1 (1347.1)
18. 81.2 (1428.3)
19. 77.4 (1505.7)
20. 80.8 (1586.5)

Wait, that can't be right. Wait, let's use a better approach. Let's list all the numbers:

77.4, 73.5, 81, 83.2, 79, 77.8, 84.1, 76, 75.8, 82.1, 80.4, 77.1, 78.8, 79.2, 81.2, 75.4, 85.1, 81.2, 77.4, 80.8

Let's group them to make addition easier:

- 77.4 appears twice: \(77.4\times2 = 154.8\)
- 81.2 appears twice: \(81.2\times2 = 162.4\)
- 73.5, 81, 83.2, 79, 77.8, 84.1, 76, 75.8, 82.1, 80.4, 77.1, 78.8, 79.2, 75.4, 85.1, 80.8

Now sum the remaining:

73.5 + 81 = 154.5

154.5 + 83.2 = 237.7

237.7 + 79 = 316.7

316.7 + 77.8 = 394.5

394.5 + 84.1 = 478.6

478.6 + 76 = 554.6

554.6 + 75.8 = 630.4

630.4 + 82.1 = 712.5

712.5 + 80.4 = 792.9

792.9 + 77.1 = 870

870 + 78.8 = 948.8

948.8 + 79.2 = 1028

1028 + 75.4 = 1103.4

1103.4 + 85.1 = 1188.5

1188.5 + 80.8 = 1269.3

Now add the grouped ones: 154.8 (from 77.4×2) + 162.4 (from 81.2×2) + 1269.3 = 154.8 + 162.4 = 317.2; 317.2 + 1269.3 = 1586.5. Wait, so total sum is 1586.5? Wait, no, wait the number of terms is 20. Wait, let's count the numbers:

1. 77.4

2. 73.5

3. 81

4. 83.2

5. 79

6. 77.8

7. 84.1

8. 76

9. 75.8

10. 82.1

11. 80.4

12. 77.1

13. 78.8

14. 79.2

15. 81.2

16. 75.4

17. 85.1

18. 81.2

19. 77.4

20. 80.8

Yes, 20 terms. So sum is 1586.5? Wait, let's check with another method. Let's use a calculator-like approach:

77.4 + 73.5 = 150.9

150.9 + 81 = 231.9

231.9 + 83.2 = 315.1

315.1 + 79 = 394.1

394.1 + 77.8 = 471.9

471.9 + 84.1 = 556

556 + 76 = 632

632 + 75.8 = 707.8

707.8 + 82.1 = 789.9

789.9 + 80.4 = 870.3

870.3 + 77.1 = 947.4

947.4 + 78.8 = 1026.2

1026.2 + 79.2 = 1105.4

1105.4 + 81.2 = 1186.6

1186.6 + 75.4 = 1262

1262 + 85.1 = 1347.1

1347.1 + 81.2 = 1428.3

1428.3 + 77.4 = 1505.7

1505.7 + 80.8 = 1586.5. Yes, so total sum is 1586.5.

Step2: Calculate the mean

The mean is the total sum divided by the number of samples (which is 20, since there are 20 sample means). So:

\[ \text{Mean} = \frac{1586.5}{20} = 79.325 \]

Wait, wait, 1586.5 divided by 20: 1586.5 ÷ 20 = 79.325. Let's check: 20 × 79 = 1580, 20 × 0.325 = 6.5, so 1580 + 6.5 = 1586.5. Correct.

Answer:

79.325