Answer to Question #337451 in Statistics and Probability for katrichie

Question

The amount of time devoted to preparing for a statistics examination by students is a normally distributed random variable with a mean of 17 hours and a standard deviation of 5 hours.

Required:

a) What is the amount of time below which only 15% of all students spend studying?

b) What is the amount of time above which only one third of all students spend studying? c) What is the probability that a student spends between 16 and 20 hours studying?

d) What is the probability that a student spends at least 15 hours studying?

e) What is the probability that a student spends at most 18 hours studying?

Answer 1

a)

$$P(Z<\dfrac{x-17}{5})=0.15$$

$$\dfrac{x-17}{5}\approx-1.0364$$

$$x\approx11.818$$

b)

$$P(Z<\dfrac{x-17}{5})=1/3$$

$$\dfrac{x-17}{5}\approx-0.4307$$

$$x\approx14.8465$$

c)

$$P(16<X<20)=P(Z<\dfrac{20-17}{5})$$

$$=P(Z<0.6)-P(Z\le-0.2)$$

$$\approx0.72575-0.42074\approx0.3050$$

d)

$$P(X\ge15)=1-P(Z<\dfrac{15-17}{5})$$

$$=1-P(Z<-0.4)$$

$$\approx0.6554$$

e)

$$P(X\le18)=P(Z\le\dfrac{18-17}{5})$$

$$=P(Z\le0.2)\approx0.5793$$