A psychologist conducts a study to determine whether or not noise can inhibit learning. Each of six subjects is tested under three experimental conditions. In each of the experimental conditions a subject is given 20 minutes to memorize a list of 10 nonsense syllables, which the subject is told she will be tested on the following day. The three experimental connditions each subject serves under are as follows: Condition 1, the no noise condition, requires subjects to study the list of nonsense syllables in a quiet room. Condition 2, the moderate noise condition, requires subjects to study the list of nonsense syllables while listening to classical music. Condition 3, the extreme noise condition, requires subjects to study the list of nonsense syllables while listening to rock music. Although in each of the experimental conditions subjects are presented with a different list of nonsense syllables, the three lists are comparable with respect to those variables that are known to influence a person’s ability to learn nonsense syllables. To control for order effects, the order of presentation of the three experimental conditions is completely counterbalanced. The number of nonsense syllables correctly recalled by the six subjects under the three experimental conditions follows: (Subjects’ scores are listed in the order Condition 1, Condition 2, Condition 3.)

library(tidyverse)
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library(ggpubr)
library(rstatix)

Attaching package: 'rstatix'

The following object is masked from 'package:stats':

    filter
library(readxl)
Friedman <- read_excel("D:/MARV BS MATH/4th year, 2nd sem/Nonparametric Statistics/Final Exam/Friedman.xlsx")
Friedman
# A tibble: 6 × 4
  Subject Condition1 Condition2 Condition3
    <dbl>      <dbl>      <dbl>      <dbl>
1       1          9          7          4
2       2         10          8          7
3       3          7          5          3
4       4         10          8          7
5       5          7          5          2
6       6          8          6          6
ks.test(Friedman,"pnorm")
Warning in ks.test.default(Friedman, "pnorm"): ties should not be present for
the Kolmogorov-Smirnov test

    Asymptotic one-sample Kolmogorov-Smirnov test

data:  Friedman
D = 0.93558, p-value < 2.2e-16
alternative hypothesis: two-sided

Since the p-value 2.2e-16 is less than D = 0.84134, then we reject the null hypothesis and conclude that at least one value does not match the specifies distribution.

Null Hypothesis: The mean response of nonsense syllables correctly recalled by the six subjects under the three experimental conditions is the same for the three condition.

Alternative Hypothesis: The mean response of nonsense syllables correctly recalled by the six subjects under the three experimental conditions is not the same for the three condition.

F <- Friedman %>%
  gather(key = "Condition", value = "score", Condition1, Condition2, Condition3) %>%
  convert_as_factor(Subject, Condition)
head(F, 6)
# A tibble: 6 × 3
  Subject Condition  score
  <fct>   <fct>      <dbl>
1 1       Condition1     9
2 2       Condition1    10
3 3       Condition1     7
4 4       Condition1    10
5 5       Condition1     7
6 6       Condition1     8
summary(Friedman)
    Subject       Condition1      Condition2     Condition3   
 Min.   :1.00   Min.   : 7.00   Min.   :5.00   Min.   :2.000  
 1st Qu.:2.25   1st Qu.: 7.25   1st Qu.:5.25   1st Qu.:3.250  
 Median :3.50   Median : 8.50   Median :6.50   Median :5.000  
 Mean   :3.50   Mean   : 8.50   Mean   :6.50   Mean   :4.833  
 3rd Qu.:4.75   3rd Qu.: 9.75   3rd Qu.:7.75   3rd Qu.:6.750  
 Max.   :6.00   Max.   :10.00   Max.   :8.00   Max.   :7.000  
ggboxplot(F, x = "Condition", y = "score", add = "jitter")

res.fried <- F %>% friedman_test(score ~ Condition |Subject)
res.fried
## # A tibble: 1 × 6
##   .y.       n statistic    df       p method       
## * <chr> <int>     <dbl> <dbl>   <dbl> <chr>        
## 1 score     6      11.6     2 0.00308 Friedman test

There was a statistically significant difference in inhibiting learning depending on which type of noise was upon the memorizing of nonsense syllable of the subject , χ2 = 11.56522, p = 0.003080668

F %>% friedman_effsize(score ~ Condition |Subject)
# A tibble: 1 × 5
  .y.       n effsize method    magnitude
* <chr> <int>   <dbl> <chr>     <ord>    
1 score     6   0.964 Kendall W large

A large effect size is detected, W = 0.9637681.

# pairwise comparisons
pwc <- F %>%
  wilcox_test(score ~ Condition, paired = TRUE, p.adjust.method = "bonferroni")
pwc
# A tibble: 3 × 9
  .y.   group1     group2        n1    n2 statistic     p p.adj p.adj.signif
* <chr> <chr>      <chr>      <int> <int>     <dbl> <dbl> <dbl> <chr>       
1 score Condition1 Condition2     6     6        21 0.02  0.059 ns          
2 score Condition1 Condition3     6     6        21 0.035 0.105 ns          
3 score Condition2 Condition3     6     6        15 0.057 0.17  ns

Condition1 and Condition2 are not statistically significant with p-value 0.059.

Condition1 and Condition3 are not statistically significant with p-value 0.105.

Condition2 and Condition3 are not statistically significant with p-value 0.170.

# Visualization: box plots with p-values
pwc <- pwc %>% add_xy_position(x = "Condition")
ggboxplot(F, x = "Condition", y = "score", add = "point") +
  stat_pvalue_manual(pwc, hide.ns = TRUE) +
  labs(
    subtitle = get_test_label(res.fried,  detailed = TRUE),
    caption = get_pwc_label(pwc)
  )

Do the data indicate that noise influenced subjects’ performance?

By the result presented above, the answer is affirmative.