Finding the probability of students learning material within a certain timeframe

Question

I am trying to determine the probability of students learning a set of instructions within a given period of time. The length of time it takes for the students to learn has a normal distribution with a mean of 24min and a standard deviation of 8min. There is only one copy of the instructions, so as soon as one person has completed the learning process, the next person begins, with no intervening gap. What is the probability that the total time for 25 learners to complete the process will be less than 9 hours (i.e. 540 minutes)?

I first wrote the probability for a single learner:

540/25=21.6min

p(x<21.6) = p((x-μ)/(σ/√n)<(21.6-24)/(8/√25)

Using normal approximation:

p(z<-1.5)

=p(z>1.5)

=0.066807

Does this check out?

Answer 1

If $X_i,$ for $i =1,2, \dots, 25,$ are a random sample from $\mathsf{Norm}(\mu = 24, \sigma = 8),$ then $$S = \sum_{n=1}^{25} X_i \sim \mathsf{Norm}(25\mu = 600, \sqrt{25\sigma^2}= 40),$$ by independence.

You seek $P(S < 540) = 0.1587.$ Computation in R below or by standardizing and using printed normal CDF tables.

pnorm(540, 600, 40)
[1] 0.0668072

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A simulation gives an approximate verification and a figure. [Using 100,000 iterations gives about two or three significant digits of accuracy.]

set.seed(2020)
s = replicate(10^5, sum(rnorm(25, 24, 8)))
mean(s); sd(s)
[1] 599.9223   # aprx E(S) = 600
[1] 39.87919   # aprx SD(S) = 40
mean(s < 540)   
[1] 0.06765    # aprx P(S < 540) = 0.0668

hist(s, prob=T, br=30, col="skyblue2",
     main="Simulated Totals")
 curve(dnorm(x,600,40), add=T, col="orange", lwd=2)
 abline(v = 540, lty="dotted", lwd=3, col="darkgreen")