How productive are U.S. workers? One way to answer this question is to study annual profits per employee. A random sample of companies in computers (I), aerospace (II), heavy equipment (III), and broadcasting (IV) gave the following data regarding annual profits per employee (units in thousands of dollars).1 computers 27.4 2 computers 23.3 3 computers 14.3 4 computers 8.2 5 computers 11.7 6 aerospace 13.8 7 aerospace 9.7 8 aerospace 11.9 9 aerospace 8.8 10 aerospace 6.2 11 aerospace 19.4 12 heavy equipment 22.2 13 heavy equipment 20.3 14 heavy equipment 7.7 15 heavy equipment 12.6 16 heavy equipment 7.5 17 broadcasting 17.2 18 broadcasting 16.8 19 broadcasting 14.1 20 broadcasting 15.8 21 broadcasting 10.4 22 broadcasting 9.5 Find SSTOT. (Round your answer to three decimal places.) SSTOT =____.


To find \( \text{SSTOT} \) (the total sum of squares), we need to follow these steps:

1. **Calculate the overall mean** \( \bar{X} \) of all the data points.
2. **Compute the total sum of squares** \( \text{SSTOT} \), which is the sum of the squared differences between each data point and the overall mean.

Let's break this down.

### Step 1: Calculate the overall mean (\( \bar{X} \))

We are given 22 data points for profits per employee (in thousands of dollars):

- Computers: 27.4, 23.3, 14.3, 8.2, 11.7
- Aerospace: 13.8, 9.7, 11.9, 8.8, 6.2, 19.4
- Heavy Equipment: 22.2, 20.3, 7.7, 12.6, 7.5
- Broadcasting: 17.2, 16.8, 14.1, 15.8, 10.4, 9.5

First, we calculate the sum of all data points:

\[
\text{Sum of all values} = 27.4 + 23.3 + 14.3 + 8.2 + 11.7 + 13.8 + 9.7 + 11.9 + 8.8 + 6.2 + 19.4 + 22.2 + 20.3 + 7.7 + 12.6 + 7.5 + 17.2 + 16.8 + 14.1 + 15.8 + 10.4 + 9.5
\]

Summing these values gives:

\[
\text{Sum} = 308.9
\]

Now, we calculate the overall mean \( \bar{X} \):

\[
\bar{X} = \frac{308.9}{22} = 14.041
\]

### Step 2: Compute SSTOT

The formula for SSTOT is:

\[
\text{SSTOT} = \sum (X_i - \bar{X})^2
\]

Where \( X_i \) is each individual data point, and \( \bar{X} = 14.041 \).

We now subtract \( \bar{X} \) from each data point, square the result, and sum them all:

\[
\text{SSTOT} = (27.4 - 14.041)^2 + (23.3 - 14.041)^2 + \dots + (9.5 - 14.041)^2
\]

Let's calculate each term:

- \( (27.4 - 14.041)^2 = 176.93 \)
- \( (23.3 - 14.041)^2 = 85.53 \)
- \( (14.3 - 14.041)^2 = 0.07 \)
- \( (8.2 - 14.041)^2 = 34.11 \)
- \( (11.7 - 14.041)^2 = 5.49 \)
- \( (13.8 - 14.041)^2 = 0.06 \)
- \( (9.7 - 14.041)^2 = 18.79 \)
- \( (11.9 - 14.041)^2 = 4.59 \)
- \( (8.8 - 14.041)^2 = 27.44 \)
- \( (6.2 - 14.041)^2 = 61.62 \)
- \( (19.4 - 14.041)^2 = 28.67 \)
- \( (22.2 - 14.041)^2 = 66.83 \)
- \( (20.3 - 14.041)^2 = 39.27 \)
- \( (7.7 - 14.041)^2 = 40.24 \)
- \( (12.6 - 14.041)^2 = 2.08 \)
- \( (7.5 - 14.041)^2 = 42.88 \)
- \( (17.2 - 14.041)^2 = 9.98 \)
- \( (16.8 - 14.041)^2 = 7.58 \)
- \( (14.1 - 14.041)^2 = 0.00 \)
- \( (15.8 - 14.041)^2 = 3.09 \)
- \( (10.4 - 14.041)^2 = 13.28 \)
- \( (9.5 - 14.041)^2 = 20.73 \)

Now, summing all these squared differences:

\[
\text{SSTOT} = 708.25
\]

### Final Answer:

\[
\boxed{\text{SSTOT} = 708.250}
\]

Do you want further details or have any questions?

Here are five related questions to expand on this topic:

1. How do we interpret the SSTOT value in terms of variance?
2. What is the next step in an ANOVA calculation after finding SSTOT?
3. How is the overall mean different from the group-specific means?
4. How would removing an outlier affect the SSTOT?
5. What assumptions must hold for valid ANOVA results?

**Tip:** SSTOT helps measure total variability; higher values indicate more spread in the data.

How productive are U.S. workers? One way to answer this question is to study annual profits per employee. A random sample of companies in computers (I), aerospace (II), heavy equipment (III), and broadcasting (IV) gave the following data regarding annual profits per employee (units in thousands of dollars). 1 computers 27.4 2 computers 23.3 3 computers 14.3 4 computers 8.2 5 computers 11.7 6 aerospace 13.8 7 aerospace 9.7 8 aerospace 11.9 9 aerospace 8.8 10 aerospace 6.2 11 aerospace 19.4 12 heavy equipment 22.2 13 heavy equipment 20.3 14 heavy equipment 7.7 15 heavy equipment 12.6 16 heavy equipment 7.5 17 broadcasting 17.2 18 broadcasting 16.8 19 broadcasting 14.1 20 broadcasting 15.8 21 broadcasting 10.4 22 broadcasting 9.5 Find SSTOT. (Round your answer to three decimal places.) SSTOT =____.

Learn how to calculate the Total Sum of Squares (SSTOT) for a random sample of companies in the computers, aerospace, heavy equipment, and broadcasting sectors. This problem involves finding the overall mean and summing the squared deviations from this mean. Suitable for students studying statistics or ANOVA at the undergraduate level.

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