Answer to Question #207635 in Statistics and Probability for romi

Question

] Suppose that the Mean and S.D of the tuition fee paid by BS Mathematics students in UMT is 150 and 30 in UDs, respectively. It is assumed that data is normally distributed. If a student is selected at random, find the probability that the amount paid by him is

  i)        Less than 105

  ii)           Between 120 to 180

 iii)           Not between 140 and 160

 iv)           Greater than 180

  v)           Between 130 and 160 GIVEN THAT between 140 to 180

vi)        Between 120 and 160 OR Between 130 and 180

Answer 1

Let $X=$ the amount paid by student; $X\sim N(\mu, \sigma^2).$

Given $\mu=150, \sigma=30$

i)

$$P(X<105)=P(Z<\dfrac{105-150}{30})$$

$$=P(Z<-1.5)\approx0.0668$$

ii)

$$=P(120<X<180)$$

$$=P(X<180)-P(X\leq120)$$

$$=P(Z<\dfrac{180-150}{30})-P(Z\leq\dfrac{120-150}{30})$$

$$=P(Z<1)-P(Z\leq-1)$$

$$\approx0.841345-0.158655\approx0.6827$$

iii)

$$P(X<140\ or\ X>160)$$

$$=P(X<140)+1-P(X\leq160)$$

$$=P(Z<\dfrac{140-150}{30})+1-P(Z\leq\dfrac{160-150}{30})$$

$$\approx P(Z<-0.33333)+1-P(Z\leq0.33333)$$

$$\approx0.36944+0.36944\approx0.7389$$

iv)

$$P(X>180)=1-P(X\leq180)$$

$$=1-P(Z\leq \dfrac{180-150}{30})$$

$$=1-P(Z\leq1)\approx0.1587$$

v)

$$P(130<X<160|140<X<180)$$

$$=\dfrac{P(130<X<160\cap 140<X<180)}{P(140<X<180)}$$

$$=\dfrac{P(140<X<160)}{P(140<X<180)}$$

$$=\dfrac{P(Z<\dfrac{160-150}{30})-P(Z\leq\dfrac{140-150}{30})}{P(Z<\dfrac{180-150}{30})-P(Z\leq\dfrac{140-150}{30}))}$$

$$\approx\dfrac{P(Z<0.33333)-P(Z\leq-0.33333)}{P(Z<1)-P(Z\leq-0.33333)}$$

$$\approx\dfrac{0.63056-0.36944}{0.84134-0.36944}\approx0.5533$$

vi)

$$P(120<X<160\ or\ 130<X<180)$$

$$=P(120<X<180)$$

$$=P(X<180)-P(X\leq120)$$

$$=P(Z<\dfrac{180-150}{30})-P(Z\leq\dfrac{120-150}{30})$$

$$=P(Z<1)-P(Z\leq-1)$$

$$\approx0.841345-0.158655\approx0.6827$$