Setup: Load Packages and Import Data

# Install negligible if needed (uncomment lines below if not installed)
# install.packages("negligible")
# install.packages("effectsize")
# install.packages("dplyr")

library(negligible)
library(effectsize)
library(dplyr)
# Import data and name it perfectionism
perfectionism <- read.csv("Assignment2_data.csv")

# Alternatively, if the CSV is unavailable:
# library(negligible)
# data(perfectionism)

# Preview the data
head(perfectionism)
##       group mpshfpre.sop mpshfpre.spp pcipre.total baipre.total cesdpre.total
## 1 CBT Group           89           57           78           13            35
## 2 CBT Group           83           70           52           21            37
## 3 CBT Group           89           76           88           33            31
## 4 CBT Group           89           57           78           13            35
## 5 CBT Group           89           59           57            9            19
## 6 CBT Group           83           79           43            8            20
##   mpsfpre.cm mpshfpost.sop mpshfpost.spp pcipost.total baipost.total
## 1         30            82            50      57.00000            15
## 2         33            47            30      34.00000            10
## 3         30            64            71      67.00000            16
## 4         30            82            50      57.00000            15
## 5         31            70            57      57.04298             6
## 6         33            78            60      26.00000             4
##   cesdpost.total mpsfpost.cm atqpre.total atqpost.total mpshfpre.oop
## 1             32          28           93            73           74
## 2             19          18           66            42           74
## 3             12          26           88            69           68
## 4             32          28           93            73           74
## 5             24          23           58            44           65
## 6             14          23           82            51           75
##   mpshfpost.oop
## 1            69
## 2            42
## 3            66
## 4            69
## 5            65
## 6            67
# Subset to CBT group only
cbt <- perfectionism %>% filter(group == "CBT Group")

# Confirm group and variable of interest
nrow(cbt)  # should be 30
## [1] 30
summary(cbt$mpshfpost.spp)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   30.00   50.00   55.50   55.99   63.00   73.00

Part a) Normality Assumption

# Shapiro-Wilk test for normality
shapiro.test(cbt$mpshfpost.spp)
## 
##  Shapiro-Wilk normality test
## 
## data:  cbt$mpshfpost.spp
## W = 0.96944, p-value = 0.5239
# QQ plot for visual inspection
qqnorm(cbt$mpshfpost.spp, main = "Q-Q Plot: SPP Scores After CBT")
qqline(cbt$mpshfpost.spp, col = "steelblue", lwd = 2)

# Histogram
hist(cbt$mpshfpost.spp,
     main = "Histogram: SPP Scores After CBT",
     xlab = "SPP Post Score",
     col = "lightblue",
     border = "white")

Interpretation:
If the p-value is greater than α = .10, we fail to reject the null hypothesis of normality, meaning the normality assumption is met. We also examine the plot: if the points fall approximately along the line, normality is supported. Based on these results, we conclude normality is met.

Part b) Effect Sizes

# The clinical threshold (mu) is 60
mu <- 60

# Cohen's d for one-sample (distance from the mean to the threshold, scaled by SD)
# Cohen's d = (mean - mu) / SD
mean_spp <- mean(cbt$mpshfpost.spp)
sd_spp   <- sd(cbt$mpshfpost.spp)
n_cbt    <- nrow(cbt)

cohens_d <- (mean_spp - mu) / sd_spp
cat("Cohen's d:", round(cohens_d, 3), "\n")
## Cohen's d: -0.366
# Interpret: negative d means the mean is below the threshold (desired direction)
# |d|: small = .2-.5, medium = .5-.8, large = .8+

Interpretation:
Cohen’s d = −0.367, indicating a small effect. Post-CBT SPP scores fell below the clinical threshold of 60, suggesting CBT produced a small but meaningful reduction in socially-prescribed perfectionism.

Part c) Mean and 90% Confidence Interval

# One-sample t-test (two-sided) at 90% CI level
# Note: t.test with conf.level = 0.90 gives a 90% CI for the mean
ci_result <- t.test(cbt$mpshfpost.spp, mu = 60, conf.level = 0.90)

cat("Mean SPP after CBT:", round(mean(cbt$mpshfpost.spp), 3), "\n")
## Mean SPP after CBT: 55.987
cat("90% Confidence Interval: [", 
    round(ci_result$conf.int[1], 3), ",", 
    round(ci_result$conf.int[2], 3), "]\n")
## 90% Confidence Interval: [ 52.59 , 59.384 ]

Interpretation:
The mean SPP score after CBT was M = 55.99, 90% CI [52.59, 59.38]. The confidence interval has a width of approximately 6.79 points, which is relatively narrow, indicating reasonably good precision in estimating the population mean.

Part d) Null Hypothesis Significance Testing (NHST)

# One-tailed (one-sided) t-test: testing if spp post-CBT < 60
# H0: mu >= 60
# H1: mu < 60 (one-tailed, lower tail)

nhst_result <- t.test(cbt$mpshfpost.spp,
                      mu = 60,
                      alternative = "less",   # directional: scores < 60
                      conf.level = 0.90)

print(nhst_result)
## 
##  One Sample t-test
## 
## data:  cbt$mpshfpost.spp
## t = -2.0073, df = 29, p-value = 0.02706
## alternative hypothesis: true mean is less than 60
## 90 percent confidence interval:
##      -Inf 58.60888
## sample estimates:
## mean of x 
##  55.98711

Hypotheses:

Decision Rule:
Using α = .10 and a one-tailed test, we reject H₀ if p < .10.

Results:
t(29) = -2.007, p = 0.027.

Decision:
Since p = .027 < α = .10, we reject H₀. There is sufficient evidence to conclude that the mean SPP score after CBT is significantly less than the clinical cut-off of 60. ```