library(graphics)
library(ggplot2)
Warning: package 'ggplot2' was built under R version 4.2.2
library(readxl)
Kolmogorov <- read_excel("D:/MARV BS MATH/4th year, 2nd sem/Nonparametric Statistics/Kolmogorov.xlsx")
Kolmogorov
# A tibble: 30 × 2
   Patient Scores
     <dbl>  <dbl>
 1       1     21
 2       2     32
 3       3     38
 4       4     40
 5       5     48
 6       6     55
 7       7     63
 8       8     66
 9       9     70
10      10     75
# … with 20 more rows

Null Hypothesis : The data comes form the specified distribution.

Alternative Hypothesis: At least one value does not match the specifies distribution.

mean(Kolmogorov$Scores)
[1] 90.06667
sd(Kolmogorov$Scores)
[1] 34.79292
ks.test(Kolmogorov,"pnorm")
Warning in ks.test.default(Kolmogorov, "pnorm"): ties should not be present for
the Kolmogorov-Smirnov test

    Asymptotic one-sample Kolmogorov-Smirnov test

data:  Kolmogorov
D = 0.96532, p-value < 2.2e-16
alternative hypothesis: two-sided

Since the p-value 2.2e-16 is less than D = 0.96532, then we reject the null hypothesis and conclude that at least one value does not match the specifies distribution.

ggplot(Kolmogorov, aes(Scores)) +
  geom_density()

ggplot(data=Kolmogorov) +  
  geom_histogram( aes(Scores, ..density..) ) +
  geom_density( aes(Scores, ..density..) ) +
  geom_rug( aes(Scores) )
Warning: The dot-dot notation (`..density..`) was deprecated in ggplot2 3.4.0.
ℹ Please use `after_stat(density)` instead.
`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

Hence, the data is not normality distributed.