Consider the following production data. Items ProducedFirst Shift (7 AM-3 PM)Second Shift (3 PM-11 PM)Third Shift (11 PM-7 AM)2/5

196

216

84

2/6

198

175

133

2/7

143

233

123

2/8

140

235

113

2/9

140

181

91

2/10

170

168

78

2/11

149

206

144

2/12

191

231

119

2/13

144

184

88

2/14

162

229

142

2/15

154

173

88

2/16

161

191

131 What is the value for the degrees of freedom for the total sum of squares?

To find the degrees of freedom for the **total sum of squares** (SST) in a production data analysis, you first need to determine how the data is structured. Specifically, we are looking at the shifts as groups, with multiple observations for each shift.

### Steps to calculate the degrees of freedom for the total sum of squares:

1. **Number of groups (k):** In this case, the groups are the three shifts (First Shift, Second Shift, and Third Shift), so \( k = 3 \).

2. **Total number of observations (n):** You have 12 days of production data for each shift. Since there are three shifts for each day, the total number of observations is:

   \[
   n = 12 \text{ days} \times 3 \text{ shifts/day} = 36 \text{ observations in total}.
   \]

3. **Degrees of freedom for the total sum of squares (DFT):**
   The degrees of freedom for the total sum of squares is simply \( n - 1 \). This is because the total degrees of freedom represents the variation across all observations.

   \[
   \text{DFT} = n - 1 = 36 - 1 = 35.
   \]

Thus, the value for the degrees of freedom for the total sum of squares is **35**.

Would you like more details on how this connects to other types of sum of squares, or any further clarification?

### Related Questions:
1. How do you calculate the between-groups sum of squares in this scenario?
2. What is the degrees of freedom for the within-groups sum of squares?
3. How would you find the mean squares for between and within groups?
4. How do you interpret the F-statistic in the context of production shifts?
5. How does an ANOVA test help in understanding production differences between shifts?

### Tip:
The degrees of freedom are a critical part of variance analysis. They indicate how much data is available to estimate parameters and are used to compute test statistics in hypothesis testing.

What is the value for the degrees of freedom for the total sum of squares, given the production data for different shifts over 12 days?

Understand how to calculate the degrees of freedom for the total sum of squares (SST) in a shift-based production data scenario. Explore key statistical concepts such as the number of groups and total observations in an ANOVA framework. Suitable for advanced high school students and undergraduates.

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