Table 1

Final Grade in Math by Mindset Groups

Scenario:

A study by Blackwell, Trzesniewski, and Dweck (2007) measured mindset of junior high students at the beginning of 7^th grade, as well as these students' final grade in Math at the end of that year (7^th grade). Mindset was divided into three groups: those who had mostly growth mindset beliefs, those with some growth mindset and some fixed mindset beliefs (mixed), and those with mostly fixed mindset beliefs.

The raw scores for each final Math grade in 7^th grade for 10 randomly selected students from each of the three different mindset groups (N = 30) are provided in Table 1. Copy the table so that you can use it to do the Sum of Squares calculations. The sum for each group and the total is provided in the table.

Mindset	Math Grade	Minus	Squared
Fixed	50
Fixed	65
Fixed	65
Fixed	70
Fixed	75
Fixed	75
Fixed	80
Fixed	80
Fixed	80
Fixed	99
Fixed Sum:	739.00
N of Fixed:	10
Mixed	65
Mixed	70
Mixed	75
Mixed	75
Mixed	85
Mixed	85
Mixed	90
Mixed	91
Mixed	94
Mixed	98
Mixed Sum:	828.00
N of Mixed:	10
Growth	55
Growth	55
Growth	59
Growth	65
Growth	70
Growth	72
Growth	75
Growth	75
Growth	75
Growth	80
Growth Sum:	681.00
N of Growth:	10
Total Sum:	2248.00
Total N:	30

QUESTION 1. Compute the Between Groups Sum of Squares.

Include two numbers after the decimal point. 1096.07 is not the answer, that is incorrect.

QUESTION 2. Compute the Within Groups (Error) Sum of Squares. : 33.03 is not the answer for this question. That is incorrect.

QUESTION 3 : Compute the degrees for freedom for each source:

1. Between Groups df?
2. Within Groups (Error) df?
3. Total df? QUESTION 4 : Compute the Between Groups Mean Square. Include two numbers after the decimal point. Answer 584.04 is incorrect

QUESTION 5: Compute the Within Groups (Error) Mean Square.

Include two numbers after the decimal point. The answer 122.35 is incorrrect. QUESTION 6: Compute the calculated F-value.

Include two numbers after the decimal point. QUESTION 7: What is the critical value of F for this example?

Use 2 decimal points from the 0.05 row.

QUESTION 8 : Fill in the missing areas of the "statistical sentence" showing the results: F( , ) = , p 0.05

Round the F-value to only two numbers after the decimal point. QUESTION 9 : Determine the mean difference for each pair of means.

Include one number after the decimal point, and don't forget to include the negative sign!

1. What is the difference between the means for the Growth Mindset group compared to (minus) the Mixed Mindset group?
2. What is the difference between the means for the Growth Mindset group compared to (minus) the Fixed Mindset group?
3. What is the difference between the means for the Mixed Mindset group compared to (minus) the Fixed Mindset group?

To answer these questions, we need to compute the necessary sums of squares, degrees of freedom, and mean squares from the given data. Let's go through the steps systematically. Question 1: Compute the Between Groups Sum of Squares The formula for the Between Groups Sum of Squares (SSB) is: SSB =Y. ni(X; Xr)? Where: s 71, is the number of observations in group X, is the mean of group 1 X is the overall mean of all groups Question 2: Compute the Within Groups (Error) Sum of Squares The formula for the Within Groups Sum of Squares (SSW) is: SSW = Ex E,(Xij - Xi)2 Where: . Xij is the j-th observation in group i . X; is the mean of group i Question 3: Compute the degrees for freedom for each source . Between Groups degrees of freedom (df_B): dfB =k- 1 where k is the number of groups. . Within Groups (Error) degrees of freedom (df_W): dfw =N - k where / is the total number of observations. . Total degrees of freedom (df_T): dfr =N-1Question 4: Compute the Between Groups Mean Square The formula for the Between Groups Mean Square (MSB) is: MSB = %7 Step-by-Step Calculation Let's start by calculating the necessary values from the data provided. Calculate the overall mean (XT) X7 - Total Sum 2348.00 - 78.27 Total N 30 Calculate group means (X;) . Fixed Mindset: X Fixed = 730.00 10 = 73.00 . Mixed Mindset: X Mixed = 828.00 10 = 82.80 . Growth Mindset: X Growth - 790.00 10 = 79.00 Calculate SSB SSB = 10(73.00 - 78.27)2 + 10(82.80 - 78.27)2 + 10(79.00 - 78.27)2 Let's compute these: (73.00 - 78.27)2 = (-5.27)2 = 27.77 (82.80 - 78.27)2 = 4.532 = 20.52 (79.00 - 78.27)2 = 0.732 = 0.53So, SSB = 10 x 27.77 + 10 x 20.52 + 10 x 0.53 = 277.70 + 205.20 + 5.30 = 488.20 Calculate SSW Next, we'll calculate the Within Groups Sum of Squares. We need the individual deviations of each score from their respective group means. For Fixed Mindset: SSW Fixed = (50 - 73)2 + (65 - 73)2 + (65 - 73)2 +...+ (89 -73)2 For Mixed Mindset: SSWMixed = (65 - 82.8)2 + (70 - 82.8)2 + (75 - 82.8)2 +...+ (98 - 82.8)2 For Growth Mindset: SSWGrowth = (55 - 79)2 + (55 - 79)2 + (59 - 79)? +...+ (80 -79)2Let's perform these calculations: . Fixed Mindset: (50 - 73)2 = (-23)2 = 529 (65 - 73)2 = (-8)2 = 64 (65 - 73)2 = 64 (70 - 73)2 = (-3)2 =9 (75 - 73)2 = 4 (75 - 73)2 = 4 (80 - 73)2 = 49 (80 - 73)2 = 49 (89 - 73)2 = 256 Sum of these deviations: 529 + 64 + 64 + 9+ 4 + 4 + 49 + 49 + 256 = 1028. Mixed Mindset: (65 - 82.8)- = 323.04 (70 - 82.8) = 163.84 (75 - 82.8)' = 60.84 (75 - 82.8)- = 60.84 (85 - 82.8)- = 4.84 (85 - 82.8)' = 4.84 (90 - 82.8)2 = 51.84 (91 - 82.8)- = 67.24 (94 - 82.8)- = 123.24 (98 - 82.8)- = 233.64 Sum of these deviations: 323.04 + 163.84 + 60.84 + 60.84 + 4.84 + 4.84 + 51.84 + 67.24 + 123.24 + 233.64 = 1094.96 . Growth Mindset: (55 - 79)- = 576 (55 - 79)2 = 576 (59 - 79)- = 400 (65 - 79) = 196 (70 - 79) = 81 (72 - 79)- = 49 (75 - 79)- = 16 (75 - 79)2 = 16 (80 - 79) = 1 (80 - 79)2 = 1Sum of these deviations: 576 + 576 + 400 + 196 + 81 + 49 + 16+ 16 +1+1 = 1912 So, SSW = 1028 + 1094.96 + 1912 = 4034.96 Calculate Degrees of Freedom . Between Groups df: dfB=k-1=3-1=2 . Within Groups df: dfw = N - k =30 - 3 = 27 . Total df: dfr = N - 1=30 - 1 = 29 Calculate Mean Squares SSB MSB = dfB 488.20 2 = 244.10 MSW = SSW 4034.96 27 = 149.81 dfwLet's summarize the answers: 1. Between Groups Sum of Squares (SSB): 488.20 2. Within Groups Sum of Squares (SSW): 4034.96 3. Degrees of Freedom: Between Groups df: 2 Within Groups (Error) df: 27 . Total df: 29 4. Between Groups Mean Square (MSB): 244.10Question 5: Compute the Within Groups (Error) Mean Square The formula for the Within Groups Mean Square (MSW) is: MSW = SSW d fw We already have SSW = 4034.96 and dfw = 27: MSW = 4034.96 = 149.44 27 Question 6: Compute the calculated F-value The formula for the F-value is: F = - MSB MSW We already have MSB = 244.10 and MSW = 149.44: F = 244.10 149.44 = 1.63 Question 7: What is the critical value of F for this example? To find the critical value of F, we need the F-distribution table value for dfe = 2 and difw = 27 at Q = 0.05. Using an F-distribution table, we find that the critical value Fo.05,2,27 is approximately 3.35.Question 8: Fill in the missing areas of the "statistical sentence\" The statistical sentence showing the results should be: F(2,27) = 1.63,p = 0.05 Let's summarize all answers: 1. Within Groups (Error) Mean Square (MSW); 149.44 2. Calculated F-value: 1.63 3. Critical value of F: 3.35 4. Statistical sentence: F'(2,27) = 1.63,p = 0.05 To determine the mean difference for each pair of means, we simply subtract the means of one group from the means of another group. Here are the group means we calculated earlier: . Fixed Mindset Mean (XFixed): 73.0 . Mixed Mindset Mean (XMixed): 82.8 Growth Mindset Mean (X Growth): 79.0 Mean Differences 1. Difference between Growth Mindset and Mixed Mindset: X Growth - XMixed = 79.0 - 82.8 = -3.8 2. Difference between Growth Mindset and Fixed Mindset: X Growth - XFixed = 79.0 - 73.0 = 6.0 3. Difference between Mixed Mindset and Fixed Mindset: XMixed - XFixed = 82.8 - 73.0 = 9.8 Summary of Mean Differences: 1. Growth Mindset vs. Mixed Mindset: -3.8 2. Growth Mindset vs. Fixed Mindset: 6.0 3. Mixed Mindset vs. Fixed Mindset: 9.8