Using this table:
Code	Hemoglobin (grams per 100 ml)
1 (0g)	6.7	7.8	5.5	8.4	7.0	7.8	8.6	7.4	5.8	7.0
2 (5g)	9.9	8.4	10.4	9.3	10.7	11.9	7.1	6.4	8.6	10.6
3 (10g)	10.4	8.1	10.6	8.7	10.7	9.1	8.8	8.1	7.8	8.0
4 (15g)	9.3	9.3	7.2	7.8	9.3	10.2	8.7	8.6	9.3	7.2
Which estimates  μ ̂_i minimize the sum of squares ∑_(i=1)^4∑_(t=1)^10(Y_it-(μ_i ) ̂ )^2 ?. Find this minimum


To minimize the sum of squares \(\sum_{i=1}^4 \sum_{t=1}^{10} (Y_{it} - \hat{\mu}_i)^2\), where \(Y_{it}\) represents the hemoglobin levels for each code group \(i\) and each observation \(t\), we want to find the estimates \(\hat{\mu}_i\) that minimize the total squared deviation of the observations from their group means.

### Step-by-Step Solution

1. **Understanding the Problem:**
   The sum of squares is minimized when \(\hat{\mu}_i\) is the mean of the observations for each group \(i\). This is because the arithmetic mean is the value that minimizes the sum of squared deviations from it.

   So, we need to calculate the group mean \(\hat{\mu}_i\) for each code (1 to 4).

2. **Calculate the Group Means \(\hat{\mu}_i\):**

   The mean for each group \(i\) is given by:

   \[
   \hat{\mu}_i = \frac{1}{10} \sum_{t=1}^{10} Y_{it}
   \]

   Let's compute the means for each code:

   - **Code 1 (0g):** 
     \[
     \hat{\mu}_1 = \frac{6.7 + 7.8 + 5.5 + 8.4 + 7.0 + 7.8 + 8.6 + 7.4 + 5.8 + 7.0}{10}
     \]

   - **Code 2 (5g):** 
     \[
     \hat{\mu}_2 = \frac{9.9 + 8.4 + 10.4 + 9.3 + 10.7 + 11.9 + 7.1 + 6.4 + 8.6 + 10.6}{10}
     \]

   - **Code 3 (10g):** 
     \[
     \hat{\mu}_3 = \frac{10.4 + 8.1 + 10.6 + 8.7 + 10.7 + 9.1 + 8.8 + 8.1 + 7.8 + 8.0}{10}
     \]

   - **Code 4 (15g):** 
     \[
     \hat{\mu}_4 = \frac{9.3 + 9.3 + 7.2 + 7.8 + 9.3 + 10.2 + 8.7 + 8.6 + 9.3 + 7.2}{10}
     \]

Let's calculate these values.The group means \(\hat{\mu}_i\) for each code are:

- **Code 1 (0g):** \(\hat{\mu}_1 = 7.2\)
- **Code 2 (5g):** \(\hat{\mu}_2 = 9.33\)
- **Code 3 (10g):** \(\hat{\mu}_3 = 9.03\)
- **Code 4 (15g):** \(\hat{\mu}_4 = 8.69\)

### Step 3: Find the Minimum Sum of Squares

Now, we substitute these means back into the sum of squares formula:

\[
\min \sum_{i=1}^4 \sum_{t=1}^{10} (Y_{it} - \hat{\mu}_i)^2
\]

Let's compute this minimum sum of squares.The minimum sum of squares is approximately **56.47**.

This value represents the minimized total squared deviation of the hemoglobin levels from their respective group means.

Would you like to know more details about this calculation or explore a related concept?

### Related Questions:
1. How do you calculate the variance and standard deviation from these data?
2. What is the significance of minimizing the sum of squares in statistical analysis?
3. How would the sum of squares change if the data were different?
4. How can we use these means to perform an ANOVA test?
5. How does this result relate to linear regression?

**Tip:** Minimizing the sum of squares is a fundamental approach in many statistical methods, including regression analysis and ANOVA, to estimate model parameters that best fit the observed data.

Using this table:
Code	Hemoglobin (grams per 100 ml)
1 (0g)	6.7	7.8	5.5	8.4	7.0	7.8	8.6	7.4	5.8	7.0
2 (5g)	9.9	8.4	10.4	9.3	10.7	11.9	7.1	6.4	8.6	10.6
3 (10g)	10.4	8.1	10.6	8.7	10.7	9.1	8.8	8.1	7.8	8.0
4 (15g)	9.3	9.3	7.2	7.8	9.3	10.2	8.7	8.6	9.3	7.2
Which estimates μ̂_i minimize the sum of squares ∑_(i=1)^4∑_(t=1)^10(Y_it-(μ_i ) ̂ )^2 ?. Find this minimum.

Explore how to minimize the sum of squares for hemoglobin levels across different code groups by calculating the group means. This detailed statistical problem involves concepts like arithmetic mean and sum of squares, suitable for undergraduate-level students.

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