Respuesta :
Answer:
The probability of the sum of spins between 245 and 255 is 0.2534.
Step-by-step 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,
[tex]\mu{_{S}=n\mu=E(S)[/tex]
And the standard deviation of the distribution of the sum of values of X is given by,
[tex]\sigma_{S}=\sqrt{n}\sigma=SE(S)[/tex]
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)
[tex]=P(\frac{244.50-260}{12}<\frac{S-E(S)}{SE(S)}<\frac{255.50-260}{12})[/tex]
[tex]=P(-1.29<Z<-0.38)\\=P(Z<-0.38)-P(Z<-1.29)\\=0.35197-0.09853\\=0.25344\\\approx0.2534[/tex]
*Use a z-table for the probability.
Thus, the probability of the sum of spins between 245 and 255 is 0.2534.
