Use Central Limit Theorem Stat score are normally distribute

Use Central Limit Theorem:

Stat score are normally distributed with mean = 1518 and standard deviation = 325

a. if 25 stat scores are randomly selected, find the probability that they have a mean between 1550 and 1575

b. Why can the central limit thereom be used even though the sample size does not exceed 30?

Solution

a) suppose that x is a random variable represents the stat scores that follows normal distribution with mean 1518 and standard deviation 325.

we have to find p(1550<xbar<1575) equation 1)

where xbar = mean of x

we know that z= (xbar-1518)*sqaureroot(25) / 325 follows normal distribution with mean 0 and standard deviation 1

now

p[(1550-1518)*sqaureroot(25) / 325 < z < (1575-1518)*sqaureroot(25) / 325) (multiply every term of eq 1 by squareroot(25) / 325

= p(0.492308 < z < 0.876923)

= p(z<0.876923) - p(z<0.492308)

= 0.8097358 - 0.6887491    (From the standard normal table)

b) cental limit theorm states that if set of random variables are independent of each other with well defined mean and standard deviation then under certain condiions mean of variates follows asymtotic normal distribution as n teds to infinity. here n is the number of variates (random variables).

As central limit therem does not depend on the number of observation in each random variable, but it depends on the total number of random variables taken into account. Therfore if the number of observation in any ramdom varibale is less than 30 or 25, cenrtal limit theorem can still be valid on the large number of random variables.