Question

A student suspects that the length of all songs currently on her iPod are approximately normally distributed with a mean of 257 seconds and a standard deviation of 62 seconds. Report all answers with two decimal places.

a. What proportion of songs are less than 200 seconds?

b. What proportion of songs are between 210 and 320 seconds?

c. The shortest 10% of songs are shorter than what length?

Answer 1

Solution :

Given that mean μ = 257 , standard deviation σ = 62

a. => P(x < 200) = P((x - μ)/σ < (200 - 257)/62)

= P(Z < -0.92)

= 1 − P(Z < 0.92)

= 1 − 0.82

= 0.18

=> The proportion of songs are less than 200 seconds is 0.18

b. => P(210 < x < 320) = P((210 - 257)/62 < (x - μ)/σ < (320 - 257)/62)

= P(-0.76 < Z < 1.02)

= P(Z < 1.02) − P(Z < −0.76)

= 0.85 - (1 − P(Z < 0.76))

= 0.85 - (1 - 0.78)

= 0.85 - 0.22

= 0.63

c. => the area of p = 0.10 then Z-value = -1.28

=> X = μ + Z*σ

= 257 + (-1.28)*62

= 177.64

=> The length is 177.64