Statistics Chi Square Data Analysis Discussion Questions

SPSS required to complete. please see for ASSIGNMENT attachments and Discussion questions below (attachment are the guides/ example).

1. During this week’s lesson, you learned that the chi-square test is a nonparametric test. When

compared to parametric tests such as t or F, nonparametric tests tend to have less statistical

power than parametric tests. Explain why this is the case. Given this distinction, would a test

that has less statistical power increase the risk of a Type I error or a Type II error? Explain.

2. Is the chi-square goodness-of-fit test a univariate test or a bivariate test? In other words, does

it involve one variable or does it involve two variables? Explain.

3. Present a research question in an aviation context—other than the ones presented in the

guided example and graded assignment—that would be a good application of the chi-square

goodness-of-fit test. As part of your RQ include any corresponding operational definitions

and the targeted variable.

Runway Incursions
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Situational Awareness
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Miscommunication
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Distraction
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Airport Markings
Complex Taxiways
Complex Taxiways
Complex Taxiways
Complex Taxiways
Complex Taxiways
Complex Taxiways
Complex Taxiways
Complex Taxiways
Complex Taxiways
Complex Taxiways
Complex Taxiways
Complex Taxiways
Complex Taxiways
Complex Taxiways
Complex Taxiways
Complex Taxiways
Complex Taxiways
Complex Taxiways
Complex Taxiways
Complex Taxiways
Seat Preferences
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Aisle Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Window Seat
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Exit Row
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Middle Seat
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Delta Comfort Plus
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Main Cabin
Week 13
One-Way Chi-Square
Graded Assignment
Let’s assume that Delta Airlines reports the following information about seat preferences among
U.S. travelers who took a Delta flight to/from at least one domestic airport during the past 3 months:
• 30% prefer an aisle seat
• 20% prefer a window seat
• 20% prefer an exit row seat
• 10% prefer a middle seat
• 10% prefer to sit in Delta Comfort Plus
• 10% prefer to sit in the main cabin
Let’s further assume that you randomly surveyed 400 Delta passengers with similar flight
experiences and asked them to specify their seat preferences. A copy of these data is contained in
the Excel file titled “Week 13 Graded Assignment Data.” For the reader’s convenience these
data are summarized in the table below.
Seat Preference
Aisle
Window
Exit Row
Middle
Delta Comfort Plus
Main Cabin
f
125
77
90
31
42
35
A. Pre-Data Analysis
1. What is the research question and corresponding operational definitions?
2. What is the research methodology/design and why is this methodology appropriate?
3. Conduct an a priori power analysis to determine the minimum sample size needed. Is the
given data set sufficient relative to this minimum sample size? In what way do you think
the size of the given sample will impact the results?
B. Data Analysis
Using the data from the Excel file (or the table above), conduct a hypothesis test as follows:
1. Formulate the null and alternative hypotheses in words.
2. Determine the test criteria.
3. Confirm the data set is compliant with the two primary assumptions for the goodness-offit test.
4. Run the analysis and record the results, or calculate the chi-square goodness-of-fit test
statistic by hand similar to what was done in the guided example.
5. Make a decision to reject or fail to reject the null hypothesis and write a concluding
statement.
C. Post-Data Analysis
1. Determine and interpret the effect size.
2. Determine and interpret the power of the study.
3. Present at least three plausible explanations for the results.
4. Interpret the findings from a practical perspective.
Week 13
One-Way Chi-Square
This handout material supplements the information about one-way chi-square that is
presented in Chapter 5 of the assigned textbook by Wilson and Joye. You are encouraged to read
this chapter before reviewing the information contained here. You also should review the Week
13 Overview to ensure that you have a general understanding of the need for chi-square tests.
The Concept of the Chi-Square Test for Goodness of Fit
In the most general sense, the chi-square test is a statistical procedure that tests whether
sets of frequencies follow certain patterns. For example, Torres, Metscher, and Smith (2011),
examined the relationship between various human factor errors and the occurrence of runway
incursions. As part of their study, Torres et al. reviewed reports of runway incursions submitted to
the Aviation Safety Reporting System (ASRS) and NTSB between January 2005 and
March 2009. From these reports they recorded the number of runway incursions that were
attributed to various human factors errors such as situational awareness, miscommunications, and
airport markings. In the context of chi-square, each contributing factor is considered a category
and the number of incursions associated with each category represents the corresponding
frequency. These data usually are placed in a frequency distribution table similar to what we
presented in Week 2’s lesson. Table 13.1 contains a distribution table of the top five categories
from the Torres et al. study, but the frequencies are fictitious for instructional purposes.
When examined from a hypothesis test perspective, Torres et al. (2011) applied a oneway chi-square strategy to test the null hypothesis that the frequencies of “scores” falling into the
different categories do not differ significantly from the frequencies of scores that have been
hypothesized from theory or the literature. In the Torres et al.’s study, they hypothesized there
would be an even distribution among the runway incursion factors. In other words, they posited
Table 13.1
Number of Runway Incursions
per Human Factors Errors
Category
Situational Awareness
Miscommunication
Distraction
Airport Markings
Complex Taxiways
f
90
70
40
30
20
Note. N = 250.
Michael A. Gallo © 2018
Week 13: Gallo Supplement: The Concept of One-Way Chi-Square   Page 1  
Table 13.2
Observed and Expected Number of Runway Incursions per Human
Factors Errors Based on A Hypothesized Even Distribution
Category
Situational Awareness
Miscommunication
Distraction
Airport Markings
Complex Taxiways
Observed
Frequencies
(O)
Expected
Proportion
90
70
40
30
20
.2
.2  
.2  
.2  
.2  
Expected
Frequencies
(E)
.2 × 250 = 50
.2 × 250 = 50
.2 × 250 = 50
.2 × 250 = 50
.2 × 250 = 50
Note. N = 250.
that the proportion of incursions would be the same across all categories. With five categories,
this mean each category is expected to have 100% / 5 = 20%. Based on the data given in Table
13.1 and a sample size of N = 250, this means that each category is expected to have .20 × 250 =
50 incursions. Thus, to determine the expected frequencies, we multiplied the hypothesized
proportions by the total sample size. This is shown in Table 13.2. Note also from Table 13.2 the
following:
• The scores obtained from sample data are called observed frequencies, denoted O.
• The scores of the claimed or established distribution are called expected frequencies,
denoted E.
In short, the chi-square test for goodness of fit is used to determine if a frequency distribution
obtained using sample data is consistent with a claimed or established distribution. In other
words, it is used to compare observed data to a theoretical model. (Note: Recall from the Week
13 Overview that one-way chi-square also is called the chi-square goodness-of-fit test.)
The Chi-Square Distribution
The chi-square test is based on the chi-square distribution, which is illustrated in Figure
13.1. You will note that this distribution is not symmetrical, but instead its shape depends on the
degrees of freedom similar to the t and F distributions. Unlike the F distribution, though, but
similar to the t distribution, the chi-square distribution has only one degree of freedom. As df
increases, the chi-square distribution becomes more symmetrical. The values of chi-square—
denoted as χ2 —are nonnegative (≥ 0), and df = C − 1 where C = the number of categories. The
critical values of the chi-square distribution are provided in Table 13.3.
Michael A. Gallo © 2018
Week 13: Gallo Supplement: The Concept of One-Way Chi-Square   Page 2  
Relative Frequency
df = 1
df = 5
df = 9
χ2
Figure 13.1. The chi-square distribution.
From Table 13.3 note that as df increases so too does the critical value. Observe that this
pattern is the complete opposite of the other statistical tests we have discussed in which as df
increases the corresponding critical values decrease. This is because the degrees of freedom for
chi-square are independent of sample size—they are not related to the number of scores in a
sample. Instead, the degrees of freedom are related to the number of categories or possible scores
instead of sample size. For example, the degrees of freedom for the runway incursion example
are df = 5 – 1 = 4 because there are five categories.
Assumptions of Chi-Square
As a nonparametric test, chi-square has fewer assumptions than parametric tests. For
example, the chi-square test does not assume that: (a) data are measured on an interval or ratio
scale, (b) the scores form a normal distribution, and (c) the data fit any particular shape.
Furthermore, there is no homogeneity of variance assumption as there is with the independent
samples t test, and outliers usually are not a problem with nominal or ordinal data. In addition to
the data type requirement the chi-square test has two primary assumptions:
• Independence. Each “score” or observation must be independent of all the other
scores. In other words, there must be one score per participant.
• Sample size. A sufficiently large sample size is needed so that the expected frequency
of each cell is at least 5. Furthermore, all individual cell frequencies must be at least 1.
In other words, each table cell must have a frequency of 1 or more, and each cell
associated with the expected frequencies must have a frequency that is 5 or greater.
Michael A. Gallo © 2018
Week 13: Gallo Supplement: The Concept of One-Way Chi-Square   Page 3  
Table 13.3
Critical Values for Chi-Square
df
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
40
50
60
70
80
90
100
.995
–0.010
0.072
0.207
0.412
0.676
0.989
1.344
1.735
2.156
2.603
3.074
3.565
4.075
4.601
5.142
5.697
6.265
6.844
7.434
8.034
8.643
9.260
9.886
10.520
11.160
11.808
12.461
13.121
13.787
20.707
27.991
35.534
43.275
51.172
59.196
67.328
.99
–0.020
0.115
0.297
0.554
0.872
1.239
1.646
2.088
2.558
3.053
3.571
4.107
4.660
5.229
5.812
6.408
7.015
7.633
8.260
8.897
9.542
10.196
10.856
11.524
12.198
12.879
13.565
14.256
14.953
22.164
29.707
37.485
45.442
53.540
61.754
70.065
p
0
χ2
Proportion in Critical Region
.975
.95
.90
.10
.05
.025
.01
0.001 0.004
0.016
2.706 3.841
5.024
6.635
0.051 0.103
0.211
4.605 5.991
7.378
9.210
0.216 0.352
0.584
6.251 7.815
9.348 11.345
0.484 0.711
1.064
7.779 9.488 11.143 13.277
0.831 1.145
1.610
9.236 11.070 12.833 15.086
1.237 1.635
2.204 10.645 12.592 14.449 16.812
1.690 2.167
2.833 12.017 14.067 16.013 18.475
2.180 2.733
3.490 13.362 15.507 17.535 20.090
2.700 3.325
4.168 14.684 16.919 19.023 21.666
3.247 3.940
4.865 15.987 18.307 20.483 23.209
3.816 4.575
5.578 17.275 19.675 21.920 24.725
4.404 5.226
6.304 18.549 21.026 23.337 26.217
5.009 5.892
7.042 19.812 22.362 24.736 27.688
5.629 6.571
7.790 21.064 23.685 26.119 29.141
6.262 7.261
8.547 22.307 24.996 27.488 30.578
6.908 7.962
9.312 23.542 26.296 28.845 32.000
7.564 8.672 10.085 24.769 27.587 30.191 33.409
8.231 9.390 10.865 25.989 28.869 31.526 34.805
8.907 10.117 11.651 27.204 30.144 32.852 36.191
9.591 10.851 12.443 28.412 31.410 34.170 37.566
10.283 11.591 13.240 29.615 32.671 35.479 38.932
10.982 12.338 14.041 30.813 33.924 36.781 40.289
11.689 13.091 14.848 32.007 35.172 38.076 41.638
12.401 13.848 15.659 33.196 36.415 39.364 42.980
13.120 14.611 16.473 34.382 37.652 40.646 44.314
13.844 15.379 17.292 35.563 38.885 41.923 45.642
14.573 16.151 18.114 36.741 40.113 43.195 46.963
15.308 16.928 18.939 37.916 41.337 44.461 48.278
16.047 17.708 19.768 39.087 42.557 45.722 49.588
16.791 18.493 20.599 40.256 43.773 46.979 50.892
24.433 26.509 29.051 51.805 55.758 59.342 63.691
32.357 34.764 37.689 63.167 67.505 71.420 76.154
40.482 43.188 46.459 74.397 79.082 83.298 88.379
48.758 51.739 55.329 85.527 90.531 95.023 100.425
57.153 60.391 64.278 96.578 101.879 106.629 112.329
65.647 69.126 73.291 107.565 113.145 118.136 124.116
74.222 77.929 82.358 118.498 124.342 129.561 135.807
.005
7.879
10.597
12.838
14.860
16.750
18.548
20.278
21.955
23.589
25.188
26.757
28.300
29.819
31.319
32.801
34.267
35.718
37.156
38.582
39.997
41.401
42.796
44.181
45.559
46.928
48.290
49.645
50.993
52.336
53.672
66.766
79.490
91.952
104.215
116.321
128.299
140.169
Note. The proportions represent the areas to the right of the critical value. To look up an area to the
left of the critical value, subtract the proportion from 1 and then use the corresponding column.
Michael A. Gallo © 2018
Week 13: Gallo Supplement: The Concept of One-Way Chi-Square   Page 4  
The Test Statistic for the Chi-Square Goodness-of-Fit Test
Although we will use a statistical software program to calculate the chi-square test
statistic, it nevertheless is helpful to know the formula for this test statistic:
χ2 = Σ
(Oi − Ei ) 2
Ei
From the formula we: (a) calculate the difference between the observed (O) and expected (E)
frequencies for each category, (b)€square each difference and divide these squared values by the
corresponding expected frequency, and (c) add the individual quotients across all categories.
To illustrate how to apply this formula, we will use the data from Table 3.2 for the
runway incursion example by expanding the table to include columns associated with the various
parts of the formula. As shown in Table 13.4, we created three new columns: one for the
difference between the observed and expected frequencies (O – E), one for the squared
difference (O – E)2, and one for the quotient between the squared difference and expected
frequency (O – E)2 / E. We then summed the quotients. Thus, χ2 = 68.0.
The Effect Size for the Chi-Square Goodness-of-Fit Test
The effect size for the chi-square goodness-of-fit test is Cohen’s w:
w=
χ2
N
Similar to the Pearson r, Cohen’s w can vary between 0 and 1. As w approaches 1, the relationship
becomes stronger, and as w approaches 0 the relationship becomes weaker. Cohen also suggested
€
2
the following for χ : (a) ES = .10 is small, (b) ES = .30 is medium, and (c) ES = .50 is large.
Table 13.4
Observed and Expected Number of Runway Incursions per Human Factors Errors Based on A Hypothesized Even
Distribution and Configured to Follow the Chi-Square Test Statistic Formula
Category
Situational Awareness
Miscommunication
Distraction
Airport Markings
Complex Taxiways
Observed
Frequencies
(O)
Expected
Proportion
90
70
40
30
20
.2
.2  
.2  
.2  
.2  
Expected
Frequencies
(E)
.2 × 250 = 50
.2 × 250 = 50
.2 × 250 = 50
.2 × 250 = 50
.2 × 250 = 50
(O – E)
90 – 50 = 40
70 – 50 = 20
40 – 50 = -10
30 – 50 = -20
20 – 50 = -30
2
(O – E)
402 = 1600
202 = 400
-102 =€100
-202 = 400
-302 = 900
Sum (Σ)
(O − E) 2
E
1600/50 = 32
400/50 = 8
100/50 = 2
400/50 = 8
900/50 = 18
68.0
Note. N = 250.
Michael A. Gallo © 2018
Week 13: Gallo Supplement: The Concept of One-Way Chi-Square   Page 5  
Week 13
One-Way Chi-Square: A Guided Example
This handout material provides a guided example of one-way chi-square, which also is
called the chi-square goodness-of-fit test. The example is similar in structure to the other guided
examples that have been presented in previous weeks. Prior to working through this example you
are encouraged to review Chapter 5 of the assigned textbook by Wilson and Joye as well as the
Gallo supplement on one-way chi-square.
Guided Example Context
The context of this guided example is from Torres et al. (2011), which examined the
relationship between various human factors errors and runway incursions. The data set we will use
is summarized in Table 13.1 from the Gallo supplement on one-way chi-square, and is replicated
here for the reader’s convenience and labeled Table 1. A copy of the data also is given in the Excel
file, “Week 13 Guided Example Data.” As shown in Table 1, five categories of human factors
errors are being considered—situational awareness, miscommunication, distraction, airport
markings, and complex taxiways—and their corresponding frequencies are provided.
Pre-data analysis. Before we begin data collection, we first must pose the research
question, identify the correct research methodology to answer the RQ, and conduct an a priori
power analysis to determine the minimum sample size needed.
What is the RQ? The overriding research question for the current example is: “What is
the relationship between the observed frequencies of runway incursions for the targeted five
categories of human factors errors and their corresponding expected frequencies?” As noted in
Torres et al. (2011), runway incursions were defined as those involving pilot deviations or
operational errors; vehicle/pedestrian deviations were not included. Furthermore, the runway
incursions data were collected between January 2005 and March 2009.
Table 1
Number of Runway Incursions
per Human Factors Errors
Category
Situational Awareness
Miscommunication
Distraction
Airport Markings
Complex Taxiways
f
90
70
40
30
20
Note. N = 250.
Michael A. Gallo © 2018
Week 13: Gallo Guided Example: Applying One-Way Chi-Square  Page 1  
What is the research methodology? The research methodology that would best answer
this question is correlational because we are examining a relationship between two entities
(observed vs. expected frequencies).
What is the minimum sample size needed? To determine the minimum sample size, we
consult G•Power using the following parameters:
• Test family = χ2 tests.
• Statistical test = Goodness of-fit tests: Contingency tables.
• Type of power analysis = A priori: Compute required sample size—given α, power,
and effect size.
• Input parameters are: Effect size w = 0.3 (this is a medium effect), α error prob = .05,
Power = .80, and df = C – 1 = 5 – 1 = 4.
These parameters result in a minimum sample size of N = 133. A copy of the G*Power output is
given in Figure 1.
Figure 1. G•Power output for chi-square test for goodness of fit (a priori power analysis).
Michael A. Gallo © 2018
Week 13: Gallo Guided Example: Applying One-Way Chi-Square  Page 2  
Data analysis. We now direct our attention to hypothesis testing. Following is a
summary of the steps associated with the corresponding hypothesis test. Before doing so, though,
observe that the data do indeed satisfy the two assumptions for the chi-square goodness-of-fit
test: (a) there is independence of the observations, and (b) each cell for the expected frequencies
is 5 or greater with each cell in Table 3.1 having a frequency of at least 1.
Step 1: Formulate the null and alternative hypotheses.
H0: There is an even distribution of frequencies across all five human factors errors
categories. In other words: There is no significant difference between the observed
frequencies from the sample data and the expected frequencies.
H1: There is a significant difference between the observed frequencies from the sample
data and the expected frequencies.
Step 2: Determine the test criteria. The test statistic is χ2, the level of significance is α =
.05, and the boundary of the critical region is determined from Table 13.3, which was given in
the Week 13 Gallo supplement on one-way chi-square. Based on df = 4 and α = .05, the
corresponding critical χ2 value is 9.488 as illustrated in Figure 2.
α = .05
2
χ 4 = 9.488
Figure 2. Critical region for
χ 2 with df = 4 and α = .05.
Step 3: Collect data and compute sample statistics. The computation of the chi-square
test statistic was done in Table 13.4 of the Week 13 Gallo supplement on one-way chi-square.
Following is an alternative approach to the calculations given in Table 13.4.
χ2 = Σ
(Oi − Ei ) 2
(90 − 50) 2
(70 − 50) 2
(40 − 50) 2
(30 − 50) 2
(20 − 50) 2
=
+
+
+
+
Ei
50
50
50
50
50
=
€
€
(40) 2
(20) 2
(−10) 2
(−20) 2
(−30) 2
+
+
+
+
50€
50
50
50 €
€ 50
€
=
€
€
€
Michael A. Gallo © 2018
1600
400
100
400
900
+
+
+
+
50
50
50
50
50
€
€
€
= 32.0 + 8.0 + 2.0 + 8.0 + 18.0
€
€
=€68.0 €
Week 13: Gallo Guided Example: Applying One-Way Chi-Square  Page 3  
Step 4: Make a decision: Either reject or fail to reject the null hypothesis. The
calculated χ2  = 68.0 is greater than the χ2 critical boundary of 9.488 for df = 4 and α = .05, and
hence lies in the critical region. Therefore, the decision is to reject the null hypothesis and
conclude there is a significant relationship between the observed frequencies and the expected
frequencies. The number of runway incursions is not uniform across the targeted five categories
of human factors errors.
Post-data analysis. After completing the hypothesis test, we now perform various postdata analysis activities. For one-way chi-square these include determining and reporting the
corresponding effect size and power, and then discussing some plausible explanations for the
results.
What is the effect size? The effect size for the chi-square goodness-of-fit test is Cohen’s
w, which is given by the following formula:
w=
χ2
N
Applying this formula to our results:
w=
χ2
€=
N
68.0
=
250
0.272 ≈ 0.52
Based on Cohen’s guidance, this is a large effect size. As a result, the relationship between the
€
observed and expected frequencies
of runway
incursions yielded a large effect.
€
€
What is the power of this study? Recall that power refers to the probability that the effect
found in the sample truly exists in the parent population. To determine the actual power of this
study we consult G*Power again, but this time we make the following changes:
• The type of power analysis gets changed to Post hoc: Compute achieved power—given
α, sample size, and effect size.
• The input parameters are changed to reflect the actual effect size (ES = 0.52) and Total
sample size (N = 250).
When these changes are made, the power of the study is 1.0, which means there is a greater than
99% probability that the large effect found in the sample truly exists in the population.
What are some plausible explanations for the result? The results indicate there is a
statistically significant relationship between the observed and expected frequencies of runway
incursions. Of the targeted five factors, situational awareness was the most common, followed by
miscommunications and distractions, which led to either a pilot deviation or operational error. A
Michael A. Gallo © 2018
Week 13: Gallo Guided Example: Applying One-Way Chi-Square  Page 4  
plausible explanation for these findings is that these top three factors relate to the infallible
nature of human beings. It is conceivable that pilots can (and do) lose sight of the actual location
of their aircraft at an airport, particularly if an airport is undergoing construction, which can lead
to blocked signage or closed taxiways and runways. It also is conceivable that ATCs could lose
sight of the big picture by not knowing where the aircraft under their control are located at the
airport. With respect to miscommunications and distractions, it certainly is conceivable that
pilots either do not communicate their position to other pilots in the area or do not monitor this
information from other pilots due to being distracted by various conditions. A plausible
explanation for the last three factors, which are airport related, is the lack of consistency across
airports with respect to their markings, taxiways, and signage. Can you think if other plausible
explanations?
Michael A. Gallo © 2018
Week 13: Gallo Guided Example: Applying One-Way Chi-Square  Page 5