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
Shall we reject or not reject the claim that there is no difference in population mean annual profits per employee in each of the four types of companies? Use a 5% level of significance, Find SSTOT. (Round your answer to three decimal places.)
SSTOT =

In order to determine whether we should reject or not reject the claim that there is no difference in population mean annual profits per employee in each of the four types of companies (computers, aerospace, heavy equipment, and broadcasting), we can conduct a **one-way ANOVA (Analysis of Variance)** test. The first step of this test is to calculate the **Total Sum of Squares (SSTOT)**.

Here’s the step-by-step process for calculating SSTOT:

### Step 1: Organize the Data
You provided data for four categories: computers, aerospace, heavy equipment, and broadcasting. We will use the following values:

- **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

### Step 2: Calculate the Grand Mean (\(\overline{X}\))
The grand mean is the mean of all the data points combined, regardless of the group they belong to.

\[
\overline{X} = \frac{\text{sum of all data points}}{\text{total number of data points}}
\]

### Step 3: Calculate SSTOT
The formula for the **Total Sum of Squares (SSTOT)** is:

\[
SSTOT = \sum (X_{ij} - \overline{X})^2
\]

Where:
- \(X_{ij}\) represents each individual data point.
- \(\overline{X}\) is the grand mean.

Now, let’s calculate the grand mean and SSTOT.The Total Sum of Squares (SSTOT) is approximately **690.391** (rounded to three decimal places).

Would you like to continue with the rest of the ANOVA test or ask any questions?

Here are 5 related questions to expand on this:

1. How do you interpret the result of SSTOT in the context of ANOVA?
2. What is the next step after calculating SSTOT in the ANOVA process?
3. How is the between-group variation (SSB) calculated in ANOVA?
4. Can you explain the importance of the F-statistic in ANOVA?
5. How does the p-value in ANOVA relate to the level of significance?

**Tip:** In ANOVA, the higher the SSTOT, the more variation exists in the data. However, it's the comparison between the within-group and between-group variations that helps determine statistical significance.

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). Shall we reject or not reject the claim that there is no difference in population mean annual profits per employee in each of the four types of companies? Use a 5% level of significance, Find SSTOT.

Explore how to use a one-way ANOVA test to determine whether there is a significant difference in the population mean annual profits per employee across four industries: computers, aerospace, heavy equipment, and broadcasting. This solution includes the calculation of the total sum of squares (SSTOT) and step-by-step instructions, suitable for undergraduate statistics students.

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