Let X1, X1, ..., X64 be a i. i.d. random sample of size 64 from a distribution with probability density function: f(x) = 6x(1 − x), 0 < x < 1 Approximate P(0.45 < X <¯ 0.5). Hint: find µ and σ 2

The approximate value of is 0.3686.

Step-by-step explanation:

The pdf of X is:

Compute the mean as follows:

Compute the variance as follows:

According to the Central Limit Theorem if an unknown population is selected with mean μ and standard deviation σ and appropriately huge random samples (n > 30) are selected from this population with replacement, then the distribution of the sample means will be approximately normally.  

Then, the mean of the sample means is given by,

And the standard deviation of the sample means is given by,

The sample selected is of size n = 64 > 30. Thus a central limit theorem can be applied to approximate the sampling distribution of sample mean.

Compute the value of as follows:

Thus, the approximate value of is 0.3686.

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step-by-step explanation: