Answer to Question #347223 in Statistics and Probability for shiba

Question

4. The heights of 1000 students are normally distributed with a mean of 175 centimeters and a standard deviation of 7 centimeters. Assuming that the heights are recorded to the nearest half-centimeter, how many of these students would you expect to have heights

(a) less than 160.0 centimeters?

(b) between 171.5 and 182.0 centimeters inclusive?

(c) equal to 175.0 centimeters?

(d) greater than or equal to 188.0 centimeters?

Answer 1

(a)

$$P(X<160)=P(Z<\dfrac{160-175}{7})$$

$$\approx P(Z<-2.142857)\approx 0.0161$$

$0.0161(1000)=16$

(b)

$$P(171.5<X<182)=P(Z<\dfrac{182-175}{7})$$

$$-P(Z\le\dfrac{171.5-175}{7})\approx P(Z<1)$$

$$-P(Z\le-0.5)\approx 0.5328$$

$0.5328(1000)=533$

(c)

$$P(X=175.0)=0$$

(d)

$$P(X\ge188)=1-P(Z<\dfrac{188-175}{7})$$

$$\approx1- P(Z<1.857143)\approx 0.0316$$

$0.0316(1000)=32$