Question

Let the random variable X follow a normal distribution with a mean of μ and a standard deviation of σ. Let 1 be the mean of a sample of 36 observations randomly chosen from this population, and 2 be the mean of a sample of 25 observations randomly chosen from the same population.
a) How are 1 and 2 distributed? Write down the form of the density function and the corresponding parameters.
b) Evaluate the statement:
?(?−0.2?< ?̅1 < ?+0.2?)<?(?−0.2?< ?̅2 < ?+0.2?), as to whether it is true or false.

Answer 1

a)

Standard error of sample mean from sample 1 = = =

Standard error of sample mean from sample 2 = = =

By Central Limit theorem,

b)

where Z follows Standard Normal distribution

= 0.8849 - 0.1151

= 0.7698

where Z follows Standard Normal distribution

= 0.8413 - 0.1587

= 0.6826