Question

Given a normal population which has a mean of 110 and a standard deviation of 5, find the probability that a random sample of 49 has a mean between 109 and 112. Report your answer to four decimal places.

Answer 1

Answer:

0.9168

Step-by-step explanation:

From the data given:

Mean = 110

standard deviation = 5

Let consider a random sample n =49 which have a mean between 109 and 112.

The test statistics can be computed as:

$Z_1 = \dfrac{x- \bar x}{\dfrac{\sigma}{\sqrt{n}}}$

$Z_1 = \dfrac{109- 110}{\dfrac{5}{\sqrt{49}}}$

$Z_1= \dfrac{-1}{\dfrac{5}{7}}$

$Z_1$  = -1.4

$Z_2= \dfrac{x- \bar x}{\dfrac{\sigma}{\sqrt{n}}}$

$Z_2 = \dfrac{112- 110}{\dfrac{5}{\sqrt{49}}}$

$Z_2 = \dfrac{2}{\dfrac{5}{7}}$

$Z_2 =2.8$

Thus; P(109  <  $\overline x$  <  112) = P( - 1.4 < Z < 2.8)

= P(Z < 2.8) - P( Z < -1.4)

= 0.9974 - 0.0806

= 0.9168