4. SAT Scores

The SAT is a standardized test used by some colleges and universities in admission decisions. Approximately 1 million high school students took the SAT in 2021. The 2021 version of the SAT is composed of two sections: evidence-based reading and writing, and math. A perfect combined score for the SAT is 1600. A sample of SAT scores for the combined SAT is as follows:

\begin{tabular}{rrrrr}
740 & 1380 & 1260 & 940 & 1110 \\
1210 & 600 & 1080 & 910 & 960 \\
1310 & 850 & 800 & 920 & 1110 \\
820 & 1270 & 1040 & 870 & 1160 \\
1200 & 1080 & 1120 & 750 & 1040 \\
900 & 1060 & 1420 & 1180 & 1030
\end{tabular}

a. Show a frequency distribution and histogram. Begin with the first class starting at 600 and use a class width of 100.

b. Comment on the shape of the distribution.

c. What other observations can be made about the SAT scores based on the tabular and graphical summaries?

Sure, let's solve the problem step by step.

### a. Frequency Distribution and Histogram

Step 1: Create Classes and Frequency Distribution Table

We are given a class width of 100, and the first class starts at 600. Therefore, the classes will be:

- 600 - 699
- 700 - 799
- 800 - 899
- 900 - 999
- 1000 - 1099
- 1100 - 1199
- 1200 - 1299
- 1300 - 1399
- 1400 - 1499

Now, we need to count how many scores fall into each class.

Here are the scores provided:
```
740, 1380, 1260, 940, 1110, 1210, 600, 1080, 910, 960,
1310, 850, 800, 920, 1110, 820, 1270, 1040, 870, 1160,
1200, 1080, 1120, 750, 1040, 900, 1060, 1420, 1180, 1030
```

Frequency Distribution Table:

| Class | Frequency |
|-----------|-----------|
| 600 - 699 | 1 |
| 700 - 799 | 1 |
| 800 - 899 | 4 |
| 900 - 999 | 6 |
| 1000-1099 | 7 |
| 1100-1199 | 5 |
| 1200-1299 | 4 |
| 1300-1399 | 2 |
| 1400-1499 | 1 |

Step 2: Draw the Histogram

A histogram visualizes this frequency distribution. Here is a description of how to draw the histogram:

1. On the x-axis, mark the class intervals: 600-699, 700-799, etc.
2. On the y-axis, mark the frequencies.
3. Draw bars for each class interval where the height of each bar corresponds to its frequency.

### b. Comment on the Shape of the Distribution

After plotting the histogram, we can observe the general shape of the distribution. In this case, the distribution appears to be roughly symmetric but with a slight right skew, meaning that most values are clustered around the middle with more lower scores than higher scores.

### c. Observations from the Tabular and Graphical Summaries

Other Observations:

1. Central Tendency:
- Mean Score: The average score can be calculated by summing all the scores and dividing by the number of scores.
[tex]\[
\text{Mean} = \frac{\sum \text{Scores}}{N}
= \frac{(740 + 1380 + 1260 + \ldots + 1030)}{30}
= \frac{33610}{30}
≈ 1120.33
\][/tex]

- Median Score: The median is the middle score. Since there are 30 scores, the median will be the average of the 15th and 16th scores when sorted.
Sorting the scores, the 15th and 16th scores are: 1040, 1040.
[tex]\[
\text{Median} = \frac{1040 + 1040}{2} = 1040
\][/tex]

2. Spread:
- Standard Deviation: This measures the dispersion of scores around the mean. For simplicity, let's use the population standard deviation formula:
[tex]\[
\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}
\][/tex]
Without computing, we know it gives us an idea of how spread out the scores are around the mean.

3. Shape:
- The histogram's shape gives us a sense of the skewness and symmetry. As mentioned, it is roughly symmetric with a slight right skew.

4. Range:
- The range of the scores is the difference between the maximum and minimum scores:
[tex]\[
\text{Range} = 1420 - 600 = 820
\][/tex]

These steps and observations will help us understand the SAT score distribution from the given data.