I have never used the mosaic package, so the last step below is how I think it will work. It throws an error in my code because I do not have the package. The code before that shows how to rename your columns efficiently and how to reshape the data so that favstats can calculate statistics for all of the questions at once. I hope I am not leading you astray guessing at how favstats works.
DF <- data.frame(PersonType = c("A", "B", "A", "B"),
BadName1 = runif(4, 0, 5),
BadName2 = runif(4, 0, 5),
BadName3 = runif(4, 0, 5),
BadName4 = runif(4, 0, 5),
BadName5 = runif(4, 0, 5))
DF
#> PersonType BadName1 BadName2 BadName3 BadName4 BadName5
#> 1 A 4.4328198 2.5343342 4.826156 2.935417 1.2146830
#> 2 B 0.5161261 0.4782906 4.405942 2.942268 2.9444149
#> 3 A 0.3700707 2.6378352 2.928798 3.132720 0.4250004
#> 4 B 2.0910784 1.4851919 1.051851 4.807708 4.0694807
#How to rename several columns
colnames(DF)[2:6] <- paste0("Q", 1:5)
DF
#> PersonType Q1 Q2 Q3 Q4 Q5
#> 1 A 4.4328198 2.5343342 4.826156 2.935417 1.2146830
#> 2 B 0.5161261 0.4782906 4.405942 2.942268 2.9444149
#> 3 A 0.3700707 2.6378352 2.928798 3.132720 0.4250004
#> 4 B 2.0910784 1.4851919 1.051851 4.807708 4.0694807
library(tidyr)
DFtall <- pivot_longer(DF, cols=Q1:Q5, names_to = "Question")
DFtall
#> # A tibble: 20 x 3
#> PersonType Question value
#> <fct> <chr> <dbl>
#> 1 A Q1 4.43
#> 2 A Q2 2.53
#> 3 A Q3 4.83
#> 4 A Q4 2.94
#> 5 A Q5 1.21
#> 6 B Q1 0.516
#> 7 B Q2 0.478
#> 8 B Q3 4.41
#> 9 B Q4 2.94
#> 10 B Q5 2.94
#> 11 A Q1 0.370
#> 12 A Q2 2.64
#> 13 A Q3 2.93
#> 14 A Q4 3.13
#> 15 A Q5 0.425
#> 16 B Q1 2.09
#> 17 B Q2 1.49
#> 18 B Q3 1.05
#> 19 B Q4 4.81
#> 20 B Q5 4.07
Stats <- favstats(value ~ PersonType + Question, data = DFtall)
#> Error in favstats(value ~ PersonType + Question): could not find function "favstats"
Created on 2020-01-22 by the reprex package (v0.3.0)