Question: following the lesson 9 module and the last photo i sent is one will you'll answer with the big box
1. **State the problem:**
Given the final exam scores of 18 NCCT first year students: 76, 87, 84, 56, 58, 88, 61, 88, 67, 92, 81, 63, 73, 94, 76, 97, 79, 82, 83, 85, calculate the range, population variance, sample variance, population standard deviation, and sample standard deviation.
2. **Calculate the Range:**
The range is the difference between the highest and lowest values.
Highest value = 97
Lowest value = 56
$$R = 97 - 56 = 41$$
3. **Calculate the Mean ($\bar{x}$):**
Sum all scores and divide by the number of scores ($N=20$):
$$\bar{x} = \frac{76 + 87 + 84 + 56 + 58 + 88 + 61 + 88 + 67 + 92 + 81 + 63 + 73 + 94 + 76 + 97 + 79 + 82 + 83 + 85}{20}$$
Calculate sum:
$$76 + 87 + 84 + 56 + 58 + 88 + 61 + 88 + 67 + 92 + 81 + 63 + 73 + 94 + 76 + 97 + 79 + 82 + 83 + 85 = 1509$$
So,
$$\bar{x} = \frac{1509}{20} = 75.45$$
4. **Calculate each squared deviation $(x - \bar{x})^2$ and sum them:**
| Score ($x$) | $x - \bar{x}$ | $(x - \bar{x})^2$ |
|-------------|---------------|-------------------|
| 76 | 76 - 75.45 = 0.55 | $0.55^2 = 0.30$ |
| 87 | 11.55 | 133.30 |
| 84 | 8.55 | 73.10 |
| 56 | -19.45 | 378.30 |
| 58 | -17.45 | 304.50 |
| 88 | 12.55 | 157.50 |
| 61 | -14.45 | 208.90 |
| 88 | 12.55 | 157.50 |
| 67 | -8.45 | 71.40 |
| 92 | 16.55 | 273.90 |
| 81 | 5.55 | 30.80 |
| 63 | -12.45 | 155.00 |
| 73 | -2.45 | 6.00 |
| 94 | 18.55 | 344.20 |
| 76 | 0.55 | 0.30 |
| 97 | 21.55 | 464.30 |
| 79 | 3.55 | 12.60 |
| 82 | 6.55 | 42.90 |
| 83 | 7.55 | 57.00 |
| 85 | 9.55 | 91.20 |
Sum of squared deviations:
$$\sum (x - \bar{x})^2 = 0.30 + 133.30 + 73.10 + 378.30 + 304.50 + 157.50 + 208.90 + 157.50 + 71.40 + 273.90 + 30.80 + 155.00 + 6.00 + 344.20 + 0.30 + 464.30 + 12.60 + 42.90 + 57.00 + 91.20 = 2759.80$$
5. **Calculate Population Variance ($\sigma^2$):**
$$\sigma^2 = \frac{\sum (x - \bar{x})^2}{N} = \frac{2759.80}{20} = 137.99$$
6. **Calculate Sample Variance ($s^2$):**
$$s^2 = \frac{\sum (x - \bar{x})^2}{n - 1} = \frac{2759.80}{19} = 145.25$$
7. **Calculate Population Standard Deviation ($\sigma$):**
$$\sigma = \sqrt{\sigma^2} = \sqrt{137.99} = 11.75$$
8. **Calculate Sample Standard Deviation ($s$):**
$$s = \sqrt{s^2} = \sqrt{145.25} = 12.05$$
**Final answers:**
- Range = 41
- Population Variance = 137.99
- Sample Variance = 145.25
- Population Standard Deviation = 11.75
- Sample Standard Deviation = 12.05
These measures describe how spread out the exam scores are around the mean.