Question

The lengths of songs on the radio are normally distributed with a mean length of 210 seconds. If 38.2% of all songs have lengths between 194 and 226 seconds, then the standard deviation of this distribution is:

A) 16 seconds
B) 8 seconds
C) 32 seconds
D) 64 seconds
I’m suppose to construct a bell diagram for this question but don’t know how to without the standard deviation..... thx

Answer 1

Given that the lengths of songs on the radio are normally distributed with a mean length of 210 seconds, then the probability that a randomly selected song will have length between 194 and 226 seconds is given by:


`P(194\leq X\leq226)=P\left( (194-210)/(\sigma) \ \textless \ z\ \textless \ (226-210)/(\sigma) \right) \\ \\ =P\left(- (16)/(\sigma) \ \textless \ z\ \textless \ (16)/(\sigma) \right)=2P\left(z\ \textless \ (16)/(\sigma) \right)-1`

Given that the probability that all songs have lengths between 194 and 226 seconds is 38.2% = 0.382, then


`2P\left(z\ \textless \ (16)/(\sigma) \right)-1=0.382 \\ \\ \Rightarrow2P\left(z\ \textless \ (16)/(\sigma) \right)=1+0.382=1.382 \\ \\ \Rightarrow P\left(z\ \textless \ (16)/(\sigma) \right)= (1.382)/(2) =0.691 \\ \\ \Rightarrow P\left(z\ \textless \ (16)/(\sigma) \right)=P(z\ \textless \ 0.5) \\ \\ \Rightarrow (16)/(\sigma) =0.5 \\ \\ \Rightarrow\sigma= (16)/(0.5) =32`

Therefore, the standard deviation of the distribution is 32 seconds.