Download assign1a

Statistics and Experimental Design Review
APA figures and captions – Your text is full of them. Italicize the word “Figure” and
figure number (or underline if handwritten) and follow with a clear, concise description of
exactly what is shown. For example:
Figure 1. An illustration of the display used in the training phase of the experiment.
Hypothesis – A testable statement,often about the relationship between two variables.
Independent Variable – This is the variable that you, as the experimenter, can control
and is often the one that you suspect can influence the dependent variable.
Dependent Variable – This is often the central point of your research. For example, if
you are interested in developing a program to improve SAT scores. The SAT score itself is your
dependent variable and improving this score involves exploring independent variables (i.e.
hours studying, high school courses taken) that you think will influence the score.
Hypothesis Testing – The process of exploring possible links between independent and
dependent variables.
Null Hypothesis – (Often) A statement that asserts that the independent and dependent
variables are not related.
Alternative Hypothesis – A statement that asserts that the independent and dependent
variables are related.
Type I Error – Incorrectly rejecting the null hypothesis
Type II Error – Incorrectly failing to reject the null hypothesis.
Means and Standard Deviations – Core mathematical tools for testing hypotheses
Sum of Squares (SS): A measure of the total squared distances between each score in a
set of scores and the mean (M) of those scores.
SS = ΣX2 – (ΣX)2/n
Variance (s2): The sum of squares divided by the degrees of freedom (df) which, for a
single set of scores, is n-1. This is a measure of the variability, or scatter, among a set of scores.
s2 = SS/(n-1)
Standard Deviation (s): The square root of variance. This is another measure of the
scatter. We use the standard deviation (the square root of the variance) in order to “unsquare”
our units. (So we are expressing variance as seconds or meters instead of seconds squared and
meters squared.)
s = √ s2
Standard Error of the Mean (sM) : The expected scatter of means that you would get if
you took many, many samples.
sM = s/√n
Between-Groups t-test – A method for comparing the means between two sets of
numbers and determining the probability (p) that this difference is due to chance. In science,
generally, we assume that if an event is less than 5% likely to occur, the difference is probably
statistically significant.
When comparing two groups, you need to combine or “pool” the variance of the sets of
scores….
Pooled variance = sp2 = (SS1 + SS2)/(df1 + df2)
From this you can estimate the estimate the standard error of the two means
s (M1-M2) = √( sp2/n1 + sp2/n2)
The t-statistic (t) is a single number that you can use to evaluate the likelihood that your
observed difference occurred by chance. For two independent samples
t = (M1-M2)/ s (M1-M2)
This value is then evaluated by looking it up on a table (see below).
Bar Graphs and error bars – Bar graphs are the most common way to illustrate means in a
figure. The error bars are small vertical lines that extent one standard deviation (sometimes one
standard error) above the mean.
Dependent Variables on y-axis. Generally, when constructing scientific figures, the dependent
variable is plotted on the y-axis.
Pearson Correlation Coefficient (r): A method for evaluating the relationship between two
values. This number is always between -1 and +1 and the strength of the relationship is
measured by how far this value is from zero. Although there are ways to evaluate the strength of
non-linear relationships, we will only evaluate linear relationships in this course.
Calculating Pearson’s r requires the calculation of the sum of products (SP).
SP = ΣXY – (ΣX)( ΣY)/n
and
r = SP/√(SSXSSY)
Like values for t, this number can be evaluated to determine if the value of r would be expected
by chance.
A scatterplot is the most common way to represent the relationship between two variables and a
Regression Equations allow you to determine the equation of the line that best represents these
data. If the equation of the line is y = bx + a, then you can determine the equation of this line
with
b = SP/SSx
a = MY-bMX
Critical Values for t
df
1
0.10
Probability (p)
0.05
0.01
6.31
12.71
63.66
636.62
2
2.92
4.30
9.93
31.60
3
2.35
3.18
5.84
12.92
4
2.13
2.78
4.60
8.61
5
2.02
2.57
4.03
6.87
6
1.94
2.45
3.71
5.96
7
1.89
2.37
3.50
5.41
8
1.86
2.31
3.36
5.04
9
1.83
2.26
3.25
4.78
10
1.81
2.23
3.17
4.59
11
1.80
2.20
3.11
4.44
12
1.78
2.18
3.06
4.32
13
1.77
2.16
3.01
4.22
14
1.76
2.14
2.98
4.14
15
1.75
2.13
2.95
4.07
16
1.75
2.12
2.92
4.02
17
1.74
2.11
2.90
3.97
18
1.73
2.10
2.88
3.92
19
1.73
2.09
2.86
3.88
20
1.72
2.09
2.85
3.85
21
1.72
2.08
2.83
3.82
22
1.72
2.07
2.82
3.79
23
1.71
2.07
2.82
3.77
24
1.71
2.06
2.80
3.75
25
1.71
2.06
2.79
3.73
26
1.71
2.06
2.78
3.71
27
1.70
2.05
2.77
3.69
28
1.70
2.05
2.76
3.67
29
1.70
2.05
2.76
3.66
30
1.70
2.04
2.75
3.65
40
1.68
2.02
2.70
3.55
60
1.67
2.00
2.66
3.46
120
1.66
1.98
2.62
3.37
infinity
1.65
1.96
2.58
3.29
0.00
Critical Values for r
Probability (p )
0.1
0.05
0.02
0.01
df (n-2)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
35
40
45
50
60
70
80
90
100
0.99
0.90
0.81
0.73
0.67
0.62
0.58
0.55
0.52
0.50
0.48
0.46
0.44
0.43
0.41
0.40
0.39
0.38
0.37
0.36
0.35
0.34
0.34
0.33
0.32
0.32
0.31
0.31
0.30
0.30
0.28
0.26
0.24
0.23
0.21
0.20
0.18
0.17
0.16
1.00
0.95
0.88
0.81
0.75
0.71
0.67
0.63
0.60
0.58
0.55
0.53
0.51
0.50
0.48
0.47
0.46
0.44
0.43
0.42
0.41
0.40
0.40
0.39
0.38
0.37
0.37
0.36
0.36
0.35
0.33
0.30
0.29
0.27
0.25
0.23
0.22
0.21
0.20
1.00
0.98
0.93
0.88
0.83
0.79
0.75
0.72
0.69
0.66
0.63
0.61
0.59
0.57
0.56
0.54
0.53
0.52
0.50
0.49
0.48
0.47
0.46
0.45
0.45
0.44
0.43
0.42
0.42
0.41
0.38
0.36
0.34
0.32
0.30
0.27
0.26
0.24
0.23
1.00
0.99
0.96
0.92
0.87
0.83
0.80
0.77
0.74
0.71
0.68
0.66
0.64
0.62
0.61
0.59
0.58
0.56
0.55
0.54
0.53
0.52
0.51
0.50
0.49
0.48
0.47
0.46
0.46
0.45
0.42
0.39
0.37
0.35
0.33
0.30
0.28
0.27
0.25
Assignment 1
Part A - Reaction Time Experiment (15 points)
The following set of data is from one participant who performed the visual reaction time
experiment on Monday. The table shows reaction time (in seconds) for 20 trials for the Simple
and Choice conditions.
oice Go/No-Go
0.435
0.281
0.482
0.252
0.430
0.317
0.310
0.280
0.406
0.484
0.356
0.344
0.478
0.373
0.447
0.375
0.526
0.356
0.582
0.328
0.445
0.350
0.395
0.369
0.372
0.310
0.310
0.375
0.390
0.277
0.432
0.294
0.433
0.278
0.343
0.381
0.384
0.325
0.508
0.366
Simple
0.422
0.269
0.251
0.198
0.204
0.514
0.311
0.199
0.190
0.266
0.201
0.228
0.205
0.211
0.204
0.222
0.197
0.228
0.196
0.243
Choice Go/No-Go
0.435
0.281
0.482
0.252
0.430
0.317
0.310
0.280
0.406
0.484
0.356
0.344
0.478
0.373
0.447
0.375
0.526
0.356
0.582
0.328
0.445
0.350
0.395
0.369
0.372
0.310
0.310
0.375
0.390
0.277
0.432
0.294
0.433
0.278
0.343
0.381
0.384
0.325
0.508
0.366
Simple
0.422
0.269
0.251
0.198
0.204
0.514
0.311
0.199
0.190
0.266
0.201
0.228
0.205
0.211
0.204
0.222
0.197
0.228
0.196
0.243
1. What is the hypothesis for this experiment?
The mean reaction time for the Simple task will be significantly different than the
mean reaction time for the Choice task.
2. What are the dependent and independent variables?
DV: Reaction time in milliseconds IV: Task (Choice vs. Simple)
3. Calculate the mean reaction times for each condition
a. Mean for Simple reaction time condition: 0.248 seconds
b. Mean for Choice reaction time condition: 0.423 seconds
4. How long did it take this participant to decide which key to press?: 0.175 seconds
5. Perform an independent-samples t-test to see if the mean reaction time for the choice
condition is longer than the mean reaction time for the simple condition.
a. What is the value of t? t = 7.22
b. What is the probability of being wrong if you reject the null hypothesis?
p < 0.001
Use Graph paper to create a Bar Graph to summarize the differences in reaction times.
Include error bars that extend one standard deviation above the mean for each bar. Put an
APA style figure caption below the figure.
0.50
Mean Reaction Time (seconds)
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
Simple
Choice
Figure 1. Mean reaction times (+ SD) for the Simple and Choice tasks. Data are from a
single participant who performed 20 trials for each task.
Part B - Mental Rotation Experiment (15 Points)
The data below show the mean reaction times from one participant for the mental rotation task at
5 different angles of displacement.
1. What is the hypothesis for this experiment?
There is a significant relationship between angle of rotation and response time.
2. What are the dependent and independent variables?
There are two variables, but neither one can be called the dependent and
independent variable. It is believed that rotation angle is the IV and reaction
time is the DV.
3. Calculate the Pearson correlation coefficient: r = 0.918
4. What would be the probability of making a Type I error if you were to reject the null
Hypothesis? Critical value is 0.88 for 0.05 and 0.93 for 0.02 so p is between 0.05 and
0.02
5. Use Graph paper to Create a scatterplot and include the regression line of best fit.
6. Put an APA figure caption below the scatterplot.
Reaction Time (seconds)
5.0
Reaction Time = 0.0128 Angle + 1.81
4.0
3.0
2.0
1.0
0.0
0
50
100
150
200
Angle (degrees)
Figure 1. Scatterplot showing the relationship between angle of rotation and reaction time for
one participant performing the mental rotation experiment. The regression equation and line are
included