Question 1170647
To determine if there's a significant difference in protein expression among proteins and cell lines, we'll use a two-way ANOVA (Analysis of Variance).

**1. Set up the Data**

Here's the data organized:

| Cell Line | P53 | Akt | Cyclin D | mTor | GLUT |
|---|---|---|---|---|---|
| Colon | 130 | 121 | 160 | 131 | 171 |
| Pancreatic | 123 | 113 | 158 | 106 | 165 |
| Lung | 121 | 112 | 164 | 102 | 149 |

**2. Calculate the Necessary Sums and Means**

* **Overall Mean (Grand Mean):**
    * Sum of all values = 130 + 121 + 160 + ... + 149 = 1784
    * Total number of values (N) = 15
    * Grand Mean = 1784 / 15 ≈ 118.93

* **Cell Line Means:**
    * Colon Mean = (130 + 121 + 160 + 131 + 171) / 5 = 142.6
    * Pancreatic Mean = (123 + 113 + 158 + 106 + 165) / 5 = 133
    * Lung Mean = (121 + 112 + 164 + 102 + 149) / 5 = 129.6

* **Protein Means:**
    * P53 Mean = (130 + 123 + 121) / 3 = 124.67
    * Akt Mean = (121 + 113 + 112) / 3 = 115.33
    * Cyclin D Mean = (160 + 158 + 164) / 3 = 160.67
    * mTor Mean = (131 + 106 + 102) / 3 = 113
    * GLUT Mean = (171 + 165 + 149) / 3 = 161.67

**3. Calculate Sum of Squares (SS)**

* **SS Total (SST):**
    * SST = Σ(xᵢ - Grand Mean)²
    * SST = (130 - 118.93)² + (121 - 118.93)² + ... + (149 - 118.93)² ≈ 4165.73

* **SS Between Cell Lines (SS Cells):**
    * SS Cells = Σn(Cell Mean - Grand Mean)² (where n is the number of proteins per cell line, which is 5)
    * SS Cells = 5 * [(142.6 - 118.93)² + (133 - 118.93)² + (129.6 - 118.93)²] ≈ 3208.93

* **SS Between Proteins (SS Proteins):**
    * SS Proteins = Σn(Protein Mean - Grand Mean)² (where n is the number of cell lines per protein, which is 3)
    * SS Proteins = 3 * [(124.67 - 118.93)² + (115.33 - 118.93)² + (160.67 - 118.93)² + (113 - 118.93)² + (161.67 - 118.93)²] ≈ 893.33

* **SS Error (SSE):**
    * SSE = SST - SS Cells - SS Proteins
    * SSE = 4165.73 - 3208.93 - 893.33 ≈ 63.47

**4. Calculate Degrees of Freedom (df)**

* df Cells = Number of cell lines - 1 = 3 - 1 = 2
* df Proteins = Number of proteins - 1 = 5 - 1 = 4
* df Error = (Number of cell lines - 1) * (Number of proteins - 1) = 2 * 4 = 8
* df Total = Total number of values - 1 = 15 - 1 = 14

**5. Calculate Mean Squares (MS)**

* MS Cells = SS Cells / df Cells = 3208.93 / 2 ≈ 1604.47
* MS Proteins = SS Proteins / df Proteins = 893.33 / 4 ≈ 223.33
* MS Error = SSE / df Error = 63.47 / 8 ≈ 7.93

**6. Calculate F-statistics**

* F Cells = MS Cells / MS Error = 1604.47 / 7.93 ≈ 202.33
* F Proteins = MS Proteins / MS Error = 223.33 / 7.93 ≈ 28.16

**7. Determine Critical F-values**

* **α = 0.05:**
    * F_critical (2, 8) ≈ 4.46
    * F_critical (4, 8) ≈ 3.84
* **α = 0.01:**
    * F_critical (2, 8) ≈ 8.65
    * F_critical (4, 8) ≈ 7.01

**8. Make Decisions**

**(a) Proteins**

* F Proteins (28.16) > F_critical (4, 8) at both 0.05 and 0.01.
* **Conclusion:** There is a significant difference in protein expression among the proteins.

**(b) Cell Lines**

* F Cells (202.33) > F_critical (2, 8) at both 0.05 and 0.01.
* **Conclusion:** There is a significant difference in protein expression among the cell lines.