Estimate probability with the central limit theorem

Question

I have a task to estimate probability using CLT. $P(52<X<98)$, $X=\sum_{i=1}^{50}X_i$ and $X_i\in U([1,2])$ $X_i$ are independent.

So $X$ parameters are: $\sigma = \sqrt\frac{25}{6}, EX = 75$

This is what i did: $$P\Biggl(\frac{X-75}{\sqrt\frac{25}{6}}<\frac{98-75}{\sqrt\frac{25}{6}}\Biggl) = \Phi(11,3)$$ $$1-P\Biggl(\frac{X-75}{\sqrt\frac{25}{6}}<\frac{52-75}{\sqrt\frac{25}{6}}\Biggl) = 1-\Phi(-11,3)$$ $$\Phi(11,3)-(1-\Phi(-11,3)) = 1-(1-0) = 0$$ But the answer in my book is $0,9672.$ I have no clue where i did a mistake.

Answer 1

One way to check this by yourself would be by simulating the distribution of $\sum X_i$. In R (with tidyverse) this can be done by

require(tidyverse)
set.seed(1234)
mins <- 1
maxs <- 2
n <- 50
x1  <- seq(from = mins*n, to = maxs*n, length.out = 500)
replications <- 100000
df <- as_tibble(replicate(replications,sum(runif(n, mins, maxs))))


dens <- data.frame(value = x1, y = dnorm(x1, mean = n*(maxs+mins)/2, sd = sqrt(n*(maxs-mins)^2/12)))


ggplot(df, aes(x = value)) + 
  geom_histogram(aes(y = stat(count) / sum(count)), binwidth  = 1, col = "red", fill = "grey75")+
  geom_line(data=dens,aes(y=y), colour="blue")+
  labs(title = "Histogram of sum(X_i) with normal approx. density") + theme_bw(16, "serif") + 
  theme(plot.title = element_text(size = rel(1.2), vjust = 1.5))

where the plot looks like this:

Another quite easy way is to use the $\sigma$-rules of the normal distribution (e.g. on Wikipedia1 ; Wikipedia2), that is aproximatly 99.73% of the "mass" are within $\pm 3 \sigma$ around $\mu$.