Assume a sample of 40 observations is drawn from a populatio

Assume a sample of 40 observations is drawn from a population with mean 20 and variance 2. Compute the following. Mean and variance of x p(x LE 19) P(16 LE x LE 21.5)

Solution

a) variance of x-bar = variance of x / n = 2 /40 = 0.05

The central limit theorem states that if a large enough sample is taken (typically n > 30) then the sampling distribution of x¯ is approximately a normal distribution with a mean of

hence mean will be 20

b)as variance is 0.05

therefore standard deviation will be 0.223

For x = 19, the z-value z = (19-20) / 0.22 = -4.48

Hence P(x < 19) = P(z < 4.48)

c)