Question

Let a random variable X̄ represent the mean of a sample consisting of 16 observations. The sample mean equals 56 and the sample standard deviation equals 28.

I. Statistics Calculate the following:

1) Standard Error of the Mean = Answer

II. Probabilities

1) P(42 < X̄ < 56) = Answer %

2) P(X̄>=70) = Answer %

3) P(X̄<=70) = Answer %

Answer 1

I)

1)

Standard error of the mean = / sqrt(n) = 28 / sqrt(16)

= 7

II)

1)

Using central limit theorem,

P( < x) = P( Z < x - /  )

P(42< < 56) = P( < 56) - P( < 42)

= P( Z < 56 - 56 / 7) - P( Z < 42 - 56 / 7)

= P( Z < 0) - P( Z < -2)

= 0.5 - 0.0228

= 0.4772

= 47.72%

b)

P( > 70) = P( Z > 70 - 56 / 7)

= P( Z > 2)

= 0.0228

= 2.28%

c)

P( < 70) = P( Z < 70 - 56 / 7)

= P( Z < 2)

= 0.9772

= 97.72%