Question

The expected value of the sum of 100 spins is 280 points with a standard error of points. The probability that the sum of 100 spins will be between 245 and 255, inclusively, is approximately , using a Normal approximation with a continuity correction. (Answer to four decimal places.)

Answer 1

The probability of the sum of spins between 245 and 255 is 0.2534.

Explanation:

Let S = sum of 100 spins.

The expected value of the sum of 100 spins is, E (S) = 260.

The standard error of sum of 100 spins is, SE (S) = 12.

We need to compute the probability of the sum of spins between 245 and 255.

According to the Central Limit Theorem if we have a population with mean μ and standard deviation σ and we take appropriately huge random samples (n ≥ 30) from the population with replacement, then the distribution of the sum of values of X, i.e ∑X, will be approximately normally distributed.

Then, the mean of the distribution of the sum of values of X is given by,

`\mu{_(S)=n\mu=E(S)`

And the standard deviation of the distribution of the sum of values of X is given by,

`\sigma_(S)=√(n)\sigma=SE(S)`

Compute the probability of the sum of spins between 245 and 255 as follows:

Apply continuity correction as follows:

P (245 ≤ S ≤ 255) = P (245 - 0.50 < S < 255 - 0.50)

= P (244.50 < S < 255.50)

`=P((244.50-260)/(12)<(S-E(S))/(SE(S))<(255.50-260)/(12))`

`=P(-1.29<Z<-0.38)\\=P(Z<-0.38)-P(Z<-1.29)\\=0.35197-0.09853\\=0.25344\\\approx0.2534`

*Use a z-table for the probability.

Thus, the probability of the sum of spins between 245 and 255 is 0.2534.