Question

B and C

A simple random sample of size n = 12 is obtained from a population with u = 69 and o = 14. (a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities involving the sample mean? Assuming that this condition is true, describe the sampling distribution of x. (b) Assuming the normal model can be used, determine P(x<72.3). (c) Assuming the normal model can be used, determine P(x 2 70.2).

Answer 1

Solution :

Given that,

mean = $\mu$ = 69

standard deviation = $\sigma$ = 14

n = 12

The sampling distribution of mean is ,

$\mu$_{$\bar x$} = $\mu$ = 69

The sampling distribution of standard deviation is ,  

$\sigma$_{$\bar x$} = $\sigma$ / $\sqrt$ n = 14 / $\sqrt$ 12 = 4.0415

b)

P(x $\leq$ 72.3)

= P[(x - $\mu$ ) / $\sigma$ $\leq$ (72.3 - 69) /4.0415 ]

= P(z $\leq$ 0.82)

= 0.7939

probability = 0.7939

c)

P(x $\geq$ 70.2 ) = 1 - P(x  $\leq$ 70.2)

= 1 - P[(x - $\mu$ ) / $\sigma$ $\leq$ (70.2 - 69) /4.0412 ]

= 1 -  P(z $\leq$ 0.30 )  

= 1 - 0.6179 = 0.3821

probability = 0.3821