Question

Researchers interested in the perception of three-dimensional shapes on computer screens decide to investigate what components of a square figure or cube are necessary for viewers to perceive details of the shape. They vary the stimuli to include fully rendered cubes, cubes drawn with corners but incomplete sides, and cubes with missing corner information. The viewers are trained on how to detect subtle deformations in the shapes, and then their accuracy rate is measured across the three figure conditions. Accuracy is reported as a percent correct. Four participants are recruited for an intense study during which a large number of trials are required. The trials are presented in different orders for each participant using a random-numbers table to determine unique sequences. Assume p < .05.

Visual Stimulus	Average Percent Accuracy
Full cubes	97.5
Corners, incomplete sides	91.2
Missing corners	76.4

Fill in the missing information in the source table below and then answer the following questions.

Source	SS	df	s2	F
Between-groups	30.35
Subjects	14.94
Residual
Total	58.11

1. What is the F statistic?

2. What is the F critical value?

3. Calculate the effect size.

4. What is the HSD value?

5. What should you conclude?

A) Participants’ ability to accurately detect subtle shape deformations was significantly more accurate, on average, when perceiving fully rendered cubes versus cubes drawn with corners but incomplete sides and cubes with missing corner information. Also, participants’ ability to accurately detect subtle shape deformations was significantly more accurate, on average, when perceiving cubes drawn with corners but incomplete sides versus cubes with missing corner information.

B) Participants ability to accurately detect subtle shape deformations was significantly more accurate, on average, when perceiving fully rendered cubes versus cubes drawn with corners but incomplete sides. Participants, ability to accurately detect subtle shape deformations was significantly more accurate, on average, when perceiving cubes drawn with corners but incomplete sides versus cubes with missing corner information.

C) Participants ability to accurately detect subtle shape deformations was significantly more accurate, on average, when perceiving fully rendered cubes versus cubes drawn missing corner information.

Answer 1

Source	SS	df	s2	F
Between-groups	30.35	2	15.175	7.1
Subjects	14.94	3	4.98	2.33
Residual	12.82	6	2.137
Total	58.11	11

SS for Residual = SS Total - (SS Between-groups + Subjects Subjects)

= 58.11 - (30.35 + 14.94) = 12.82

Total observations, N = 4 * 3 = 12

Number of subjects, n = 4

Number of treatments, k = 3

df Total = N - 1 = 12 - 1 = 11

df Between groups = k - 1 = 3 - 1 = 2

df Subjects = n -1 = 4 - 1 = 3

df Residuals = (N - k) - (n - 1) = (12 - 3) - (4 - 1) = 6

S2 = SS / df

F = S2 / S2 for Residual

1. What is the F statistic?

F statistic for Between-groups is 7.1.

2. What is the F critical value?

df = 2, 11

Critical value of F at 0.05 significance level and df = 2, 11 is 3.98

3. Calculate the effect size.

Effect Size = SS Between-groups / SS Total = 30.35/ 58.11 = 0.52

4. What is the HSD value?

HSD =

where are the mean values

is a critical value of the studentized range for α, the number of treatments or samples r, and the within-groups degrees of freedom . We get this value from studentized range table.

is the within groups mean square from the ANOVA table and n is the sample size for each treatment.

From Anova table,

= 2.137 , = 6

n = 4, r = 3

For 95% confidence interval , α = 0.05

From studentized range table, = 4.339

So, HSD =  

5. What should you conclude?

Mean Average Percent Accuracy between Full cubes and Corners, incomplete sides = 97.5 - 91.2 = 6.3

Mean Average Percent Accuracy between Full cubes and Missing corners = 97.5 - 76.4 = 21.1

Mean Average Percent Accuracy between Corners, incomplete sides and Missing corners = 91.2 - 76.4 = 14.8

Since all mean averages are different, we conclude that

A) Participants’ ability to accurately detect subtle shape deformations was significantly more accurate, on average, when perceiving fully rendered cubes versus cubes drawn with corners but incomplete sides and cubes with missing corner information. Also, participants’ ability to accurately detect subtle shape deformations was significantly more accurate, on average, when perceiving cubes drawn with corners but incomplete sides versus cubes with missing corner information.