Question

A random sample of =49 observations is drawn from a population with a mean equal to 15 and a standard deviation equal to 14.

a.	Give the mean and standard deviation of the​ (repeated) sampling distribution x overbarx.
b.	Describe the shape of the sampling distribution of x overbarx. Does your answer depend on the sample​ size?
c.	Calculate the standard normal​ z-score corresponding to a value of x overbarxequals=12.5.
d.	Calculate the standard normal​ z-score corresponding to a value of x overbarxequals=19.

a.

mu Subscript x overbarμ=nothing

sigma Subscript x overbarσx=nothing

Answer 1

Solution :

Given that,

mean = $\mu$ = 15

standard deviation = $\sigma$ = 14

n = 49

(a)

The sampling distribution of mean and standard deviation is ,

$\mu$_{$\bar x$} = 15 and

$\sigma$_{$\bar x$} = $\sigma$ / $\sqrt$ n = 14 / $\sqrt$ 49 = 14 / 7 = 2

(b)

The shape of the sampling distribution of _{$\bar x$} is approximately normal .

(c)

$\bar x$ = 12.5

z = (($\bar x$ - $\mu$ _{$\bar x$} ) / $\sigma$ _{$\bar x$} = (12.5 - 15) / 2 = -1.25

z - score = -1.25

(d) .

$\bar x$ = 19

z = (($\bar x$ - $\mu$ _{$\bar x$} ) / $\sigma$ _{$\bar x$} = (19 - 15) / 2 = 2

z - score = 2