Recall the CLT: if X1, X2, . . . , Xn are iid (independent and identically distributed) random variables with mean µ andvariance σ2, then for large n, the sum Sn = X1 + X2 + · · · + Xn has an approximate normal distribution N (nµ, nσ2) andthe average ¯Xn = Sn/n has an approximate normal distribution N (µ, σ2/n) .1. Fifty-three percent of the residents of a city favour construction of a certain highway.(a) [5] If we assign Xi = 1 for a person in favour of the highway and Xi = 0 when they are not in favour, then forthis random variable, findmean µ = variance σ2 =(b) [5] If we choose a random sample of n = 25 residents, find the probability that at least 50 percent of them arein favour of that highway. (Assume the normal approximation using CLT.)

The Central Limit Theorem (CLT) is a statistical theory that states that given a sufficiently large sample size from a population with a finite level of variance, the mean of all samples from the same population will be approximately equal to the mean of the population.

(a) For this random variable, we assign Xi = 1 for a person in favour of the highway and Xi = 0 when they are not in favour. The mean µ and variance σ2 can be calculated as follows:

The mean µ is the expected value of Xi, which is the probability of a resident being in favour of the highway. So, µ = P(Xi=1) = 0.53.

The variance σ2 is the expected value of the squared deviation from the mean. Since Xi is a Bernoulli random variable, its variance is given by σ2 = µ(1-µ) = 0.53 * (1 - 0.53) = 0.2491.

(b) If we choose a random sample of n = 25 residents, we want to find the probability that at least 50 percent of them are in favour of the highway. This means at least 13 out of 25 residents (since 50 percent of 25 is 12.5 and we round up to the nearest whole number).

We can use the normal approximation to the binomial distribution here, thanks to the Central Limit Theorem. The mean and variance for the sum of these 25 random variables are nµ and nσ2 respectively, which are 250.53 = 13.25 and 250.2491 = 6.2275.

We standardize to Z = (X - µ) / σ, where X is the number of residents in favour, µ is the mean, and σ is the standard deviation (which is the square root of the variance). So, we want P(X >= 13) = P(Z >= (13 - 13.25) / sqrt(6.2275)) = P(Z >= -0.1) = 1 - P(Z < -0.1) = 1 - 0.4602 = 0.5398.

So, the probability that at least 50 percent of the residents are in favour of the highway is approximately 0.54 or 54 percent.