Question

A simple random sample of size n=81 is obtained from a population with u = 73 and o = 36. (a) Describe the sampling distribution of x. (b) What is P (x>77.6) ? (c) What is P (x565) ? (d) What is P (70.4 <x<81)? (a) Choose the correct description of the shape of the sampling distribution of x. O A. The distribution is skewed right. OB. The distribution is skewed left. OC. The distribution is approximately normal. D. The distribution is uniform. E. The shape of the distribution is unknown.

Answer 1

Solution :

Given that ,

a) Correct option is C

b) $\mu$ _{$\bar x$} = 73

$\sigma$_{$\bar x$} = $\sigma$ / $\sqrt$ n = 36 / $\sqrt$ 81 = 4

P($\bar x$ > 77.6) = 1 - P($\bar x$ < 77.6 )

= 1 - P[($\bar x$ - $\mu$ _{$\bar x$} ) / $\sigma$ _{$\bar x$} < (77.6 - 73) / 4]

= 1 - P(z < 1.15)   

= 1 - 0.8749

= 0.1251

c) P($\bar x$ < 65) = P(($\bar x$ - $\mu$ _{$\bar x$} ) / $\sigma$ _{$\bar x$} < (65 - 73) / 4 )

= P(z < - 2)

Using z table

= 0.0228

d) P(70.4 < < 81 )  

T

= P[(70.4 - 73) / < (
T
- $\mu$ _{$\bar x$}) / $\sigma$ _{$\bar x$} < (81 -73) / 4)]

= P( - 0.62 < Z < 2)

= P(Z < 2) - P(Z < - 0.62)

Using z table,  

= 0.9772 - 0.2676  

= 0.7096