Simple Linear Regression

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Question 1:
Simple Linear Regression, The graph for Class A shows that the test marks in Class A are normally distributed. A normal distribution is characterized by asymmetric, bell-shaped graph which is centred at the mean of the concerned variable as is the case of Class A’s graph. Also in any normal distribution, the mean, median and mode are all equal as it is in the case of Class A’s graph.
The graph for Class B shows that the distribution of marks for Class B is positively skewed. The graph is positively skewed because it has a longer tail in the positive direction from the centre. The positively skewed graph means that more students got low scores in the test in Class B. This could possibly have been because the students of Class B found the test to be too difficult or due to other particular reasons.
The graph for Class C shows that the distribution of marks for Class C is negatively skewed. The graph is negatively skewed because it has a longer tail in the negative direction from the center. The negatively skewed graph means that more students got high scores in the test in Class C. This could possibly have been because the students of Class C found the test to be too easy or due to other particular reasons.
Question 2:
(a) The research questions suggested are as follows:
1. Is there a significant difference in critical thinking skills between male and female 12 year students?
2. Is there a significant difference in critical thinking dispositions between male and female 12 year students?
3. Is there a positive correlation between critical thinking skills and critical thinking dispositions among 12 year old students?
(b) In the first two questions we will use the T – Test for Independent Means in the two – tailed version. In the last question, we will use the simple T – Test in the one – tailed version.
Question 3:
(a) We would use Simple Linear Regression to predict mathematics performance. This is because there are only two variables involved, namely ‘attitude towards mathematics’ and ‘mathematics performance’ whereby, ‘mathematics performance’ is the dependent variable while ‘attitude towards mathematics’ is the dependent variable’.
(b) The assumptions required to be met are as follows:
1. The regression model is linear in parameters.
2. Random Sampling: The sample we use for the regression is a random sample of size ‘n’ and it follows the model of the population from which it is derived.
3. The sample outcomes of the independent variable do not all have the same value.
4. Any error term in the model has an expected value of zero.
[c] These assumptions are important to be met in order to ensure that the regression carried out and the slope parameter derived are both unbiased.
(d)
 ‘attitude towards mathematics’ :
Mean = 463 / 15 = 30.867.
X X – Mean (X-Mean)^2
40 9.133333 83.41777169
32 1.133333 1.284443689
37 6.133333 37.61777369
28 -2.866667 8.217779689
35 4.133333 17.08444169
29 -1.866667 3.484445689
32 1.133333 1.284443689
27 -3.866667 14.95111369
30 -0.866667 0.751111689
31 0.133333 0.017777689
25 -5.866667 34.41778169
26 -4.866667 23.68444769
34 3.133333 9.817775689
33 2.133333 4.551109689
24 -6.866667 47.15111569
Total = 287.73
Standard Deviation =
(287.73/14)^(1/2) =
4.533.

 ‘mathematics performance’ :
Mean = 1008 / 15 = 67.2
X X – Mean (X-Mean)^2
80 12.8 163.84
56 -11.2 125.44
66 -1.2 1.44
49 -18.2 331.24
74 6.8 46.24
61 -6.2 38.44
77 9.8 96.04
72 4.8 23.04
68 0.8 0.64
65 -2.2 4.84
59 -8.2 67.24
54 -13.2 174.24
81 13.8 190.44
79 11.8 139.24
67 -0.2 0.04
Total = 1402.2
Standard Deviation =
(1402.2 / 14)^(1/2) =
10.01

Question 4:
The scores for both groups are normally distributed as shown by the symmetric bell – shaped graphs of both groups. In Group Y, the test scores are widely spread out which means that this group has a (comparatively) higher standard deviation. In Group X, the test scores are narrowly spread out which means that this group has a (comparatively) lower standard deviation.
Question 6:
The first graph shown depicts a strong, positive, linear relationship between ‘Grade Point Average’ and ‘Hours Spent Studying’. The second graph depicts a strong, negative, linear relationship between ‘Grade Point Average’ and ‘Hours of Video Games Played’. The last graph depicts that..