Question

A normally distributed population has a mean of 600 and a standard deviation of 60. a. Determine the probability that a random sample of size 25 selected from this population will have a sample mean less than 579. b. Determine the probability that a random sample of size 16 selected from the population will have a sample mean greater than or equal to 636.

Answer 1

Solution :

Given that ,

mean = $\mu$ = 600

standard deviation = $\sigma$ = 60

n = 25

$\mu$_{$\bar x$} = 600

$\sigma$_{$\bar x$} = $\sigma$ / $\sqrt$ n = 60 / $\sqrt$ 25 = 12

P($\bar x$ < 579) = P(($\bar x$ - $\mu$ _{$\bar x$} ) / $\sigma$ _{$\bar x$} < (579 - 600) / 12)

= P(z < -1.75)

= 0.0401

Probability = 0.0401

n = 16

$\sigma$_{$\bar x$} = $\sigma$ / $\sqrt$ n = 60 / $\sqrt$ 16 = 15

P($\bar x$$\geq$ 636) = 1 - P($\bar x$$\leq$ 636)

= 1 - P[($\bar x$ - $\mu$ _{$\bar x$} ) / $\sigma$ _{$\bar x$}$\leq$ (636 - 600) /15 ]

= 1 - P(z $\leq$ 2.4)

= 1 - 0.9918

= 0.0082

Probability = 0.0082