Question

9.18 Given a normal population whose mean is 50 and
whose standard deviation is 5, find the probability
that a random sample of
a. 4 has a mean between 49 and 52.
b. 16 has a mean between 49 and 52.
c. 25 has a mean between 49 and 52.

Answer 1

$Given \; that, μ=50, σ=5, then,\\ a) P(49<\bar x<52)=P(\frac{49-50}{\frac{5}{\sqrt{4}}}<Z<\frac{52-50}{\frac{5}{\sqrt{4}}})\\ =P(-0.4<Z<0.8)\\ =P(0<Z<0.4)+P(0<Z<0.8)\\ =0.1554+0.2881=0.4435\\ b) P(49<\bar x<52)=P(\frac{49-50}{\frac{5}{\sqrt{16}}}<Z<\frac{52-50}{\frac{5}{\sqrt{16}}})\\ =P(-0.8<Z<1.6)\\ =P(0<Z<0.8)+P(0<Z<1.6)\\ =0.2881+0.4452=0.7333\\ c) P(49<\bar x<52)=P(\frac{49-50}{\frac{5}{\sqrt{25}}}<Z<\frac{52-50}{\frac{5}{\sqrt{25}}})\\ =P(-1<Z<2)\\ =P(0<Z<1)+P(0<Z<2)\\ =0.3413+0.4772=0.8185\\$