Can I use Z-score with skewed sample but normal distributed population?

Question

I would like to compare birth weight of babies in the general population with the birth weights measured in my research sample (birth weight from babies delivered by mothers on a specific drug treatment). The birth weight in the general population follows a normal distribution. My own research sample is small (24 babies) and does not follow a normal distribution.

Is it possible to use the Z-score to find statistical differences in birth weight between the two groups? And if not, what analyses should I perform?

Data and equation

Mean birth weight population = 3390 gram

Mean birth weight sample = 3668 gram Standard deviation sample = 516 gram

Z-score = (3668 - 3390 ) / 516 = 0.455

Birth weight sample - male

3850

4995

3925

3430

3825

3905

Birth weight sample - female

3955

3635

3530

2940

3820

3300

2915

3340

Standard deviation sample female = 379

Standard deviation sample male = 525

Mean population birth weight female = 3293

Mean population birth weight male = 3487

qqplot for girls:

qqplot for boys:

Answer 1

A simple approach would be to bootstrap means.

• For the boys, this yields $p<0.0001$, that is, out of my 10,000 bootstrapped means, not a single one was below the boys' population mean weight.

• For the girls, I get $p=0.15$ for the comparison of the bootstrapped mean against the population mean (i.e., 15% of my bootstrapped means are below the population mean).

• In principle, you should really adjust these p-values for the multiple (two) tests you ran... but that won't make a difference, since $p=0$ will remain zero for the boys, and the nonsignificant result for the girls will remain nonsignificant.

Below is a plot of all your data, with group means and bootstrapped 95% confidence intervals for the means, and the population mean in red. If you had more data, you should really jitter the points horizontally.

Needless to say (I hope), you should be really careful about drawing conclusions from data samples this small.

R code:

boys <- c(3850, 4995, 3925, 3430, 3825, 3905)
girls <- c(3955, 3635, 3530, 2940, 3820, 3300, 2915, 3340)
population_boys <- 3487
population_girls <- 3293

library(boot)

# Boys:
set.seed(1) # for reproducibility
boot_boys <- boot(boys,function(x,i)mean(x[i]),R=10000)
boot_boys$t0	# mean
sd(boot_boys$t)   # standard error of the bootstrapped mean
ecdf(boot_boys$t)(population_boys)  # p value for one-sided test for equality with the population average

# Girls:
set.seed(1) # for reproducibility
boot_girls <- boot(girls,function(x,i)mean(x[i]),R=10000)
boot_girls$t0	# mean
sd(boot_girls$t)  # standard error of the bootstrapped mean
ecdf(boot_girls$t)(population_girls)    # p value for one-sided test for equality with the population average

opar <- par(mai=c(.5,1,.1,.1))
    plot(c(.6,2.4),range(c(boys,girls,population_boys,population_girls)),
        type="n",xlab="",ylab="",las=1,xaxt="n")
    mtext("Birth weight in grams",2,line=3.5)
    axis(1,c(1,2),c("Boys","Girls"))
    # plot boys
    points(rep(1,length(boys)),boys,pch=19)
    lines(c(0.8,0.8),quantile(boot_boys$t,c(0.025,0.975)))
	points(0.8,boot_boys$t0,pch=21,bg="white",cex=1.5,lwd=2)
    # plot girls
    points(rep(2,length(girls)),girls,pch=19)
    lines(c(2.2,2.2),quantile(boot_girls$t,c(0.025,0.975)))
	points(2.2,boot_girls$t0,pch=21,bg="white",cex=1.5,lwd=2)
    # plot population
    lines(c(.7,1.3),rep(population_boys,2),col="red")
    lines(c(1.7,2.3),rep(population_girls,2),col="red")
par(opar)