Question

In: Statistics and Probability

Suppose x is a random variable with a mean of 40 and a standard deviation of...

Suppose x is a random variable with a mean of 40 and a standard deviation of 6.5 that is not necessarily normally distributed.

(a) If random samples of size n=20 are selected, can you say the sampling distribution of the means, the (x bar) distribution, is normally distributed? why or why not?

(b) if random samples of size n=64 are selected, what can you say about the sampling distribution of the means, (x bar)? is it normally distributed? what is μ _(x bar) ? what is the standard error?

Expert Solution

a ) X is a random variable .

A random sample of size 20 is drawn from X .

Now , by Lindeberg-Levy Central Limit Theorem , suppose , is a sequence of independent and identically distributed random variables with and , finite and non-vanishing. If ., then ,

, where ,

Hence , taking the cue from Lindeberg-Levy Central Limit Theorem , for this problem , for a random sample of size 20 drawn from X , the sampling distribution of follows assymptotically a normal distribution with mean 40 and standard error

b ) Using Lindeberg-Levy central Limit theorem , we can say that if a random sample of size 64 is drawn from X , then follows assymptotically a normal distribution with mean 40 and standard error

orchestra answered 1 year ago

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