Mean differs from a value

A notebook made by Paul Amat, amat-design.com

Context and objectives

Does the population mean or median differ from a predefined value?

Research hypothesis

Statistical hypothesis

H0:

H1:

Data preparation and exploration

Starting session

library(readr)
data <- read_csv("data.csv", show_col_types = FALSE)

Defining null hypothesis mean

mu0 <- 34

Group descriptives

print("Score")

[1] "Score"

summary(data$score)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   6.36   18.75   30.25   30.31   41.45   54.78 

Difference between means

mean(data$score)-mu0 # ok jamovi

[1] -3.69145

Parametric test

Conditions

Independent observations

The independence of observations is guaranteed by the recruitment process.

Normally distributed data

Plots

par(mfrow=c(1,2))
hist(data$score, freq=FALSE)
curve(dnorm(x, mean=mean(data$score), sd=sd(data$score)), add=TRUE, col="red")

qqnorm(data$score)
qqline(data$score, col="red")

Kolmogorov-Smirnov

ks.test(data$score, "pnorm", mean(data$score), sd(data$score)) # ok in jamovi

Warning in ks.test(data$score, "pnorm", mean(data$score), sd(data$score)) :
  ties should not be present for the Kolmogorov-Smirnov test

    One-sample Kolmogorov-Smirnov test

data:  data$score
D = 0.083478, p-value = 0.1231
alternative hypothesis: two-sided

Shapiro-Wilk

shapiro.test(data$score) # ok in jamovi

    Shapiro-Wilk normality test

data:  data$score
W = 0.95916, p-value = 1.607e-05

No outliers

outliers <- function(x) {
   qinf <- quantile(x, 0.25) - (IQR(x)*1.5)
   qsup <- quantile(x, 0.75) + (IQR(x)*1.5)
   x[x < qinf | x > qsup]
}

outliers(data$score)

numeric(0)

Test

t.test(data$score, mu = mu0) # ok in jamovi

    One Sample t-test

data:  data$score
t = -4.0113, df = 199, p-value = 8.542e-05
alternative hypothesis: true mean is not equal to 34
95 percent confidence interval:
 28.49383 32.12327
sample estimates:
mean of x 
 30.30855 

Effect size

if(!require(effsize)) install.packages("effsize")

Le chargement a nécessité le package : effsize

library(effsize)
(cohentest <- cohen.d(data$score, NA, mu = mu0)) #ok jamovi

Cohen's d (single sample)

d estimate: -0.2836416 (small)
Reference mu: 34
95 percent confidence interval:
       lower        upper 
-0.563917120 -0.003366054 

Power analysis

Finding power

if(!require(pwr)) install.packages("pwr")

Le chargement a nécessité le package : pwr

library(pwr)
pwr.t.test(d = cohentest$estimate, n = length(data$score), sig.level = 0.05, type = "one.sample", alternative = "two.sided")

     One-sample t test power calculation 

              n = 200
              d = 0.2836416
      sig.level = 0.05
          power = 0.9789182
    alternative = two.sided

Finding N

pwr.t.test(d = cohentest$estimate, power = 0.8, sig.level = 0.05, type = "one.sample", alternative = "two.sided")

     One-sample t test power calculation 

              n = 99.4967
              d = 0.2836416
      sig.level = 0.05
          power = 0.8
    alternative = two.sided

Non-parametric test

Conditions

Independent observations

The independence of observations is guaranteed by the recruitment process.

N > 30

length(data$score) > 30

[1] TRUE

Test

wilcox.test(data$score, mu = mu0) # ok jamovi

    Wilcoxon signed rank test with continuity correction

data:  data$score
V = 6951, p-value = 0.0001564
alternative hypothesis: true location is not equal to 34

Effect size

if(!require(effectsize)) install.packages("effectsize")

Le chargement a nécessité le package : effectsize
Registered S3 method overwritten by 'parameters':
  method                         from      
  format.parameters_distribution datawizard

library(effectsize)

rank_biserial(data$score, mu = mu0)

r (rank biserial) |         95% CI
----------------------------------
-0.31             | [-0.45, -0.16]

- Deviation from a difference of 34.

Results and recommendations