Question

(1 point) The song-length of tunes in the Big Hair playlist of a certain Statistics professors mp3-player vary from song to song. This variation can be modeled by the Normal distribution, with a mean song-length of ?=4.1μ=4.1 minutes and a standard deviation of ?=0.6σ=0.6 minutes. Note that a song that has a length of 4.5 minutes is a song that lasts for 4 minutes and 30 seconds.

While listening to a song, the professor decides to shuffle the playlist, which means the mp3-player is to randomly pick a song within this particular playlist, and play this next.

If using/working with ?z-values, use three decimals.

(a) What is the probability that the next song to be played is between 4 and 4.6 minutes long? Answer to four decimals.

(b) What proportion of all the songs in this playlist are longer than 5 minutes? Use four decimals in your answer.

(c) 15% of all the songs in this playlist are at most how long, in minutes? Enter your answer to two decimals, and keep your answer consistent with how the song length has been expressed in this problem.

minutes

(d) There are 225 songs in the Big Hair playlist. How many of these would you expect to be longer than 5 minutes in length? Use two decimals in your answer.

songs

(e) From the time he set his mp3-player to shuffle, there has been 17 songs randomly chosen and played in succession. What is the chance that the 17-th song played is the 8-th to be longer than 4.1 minutes? Enter your answer to four decimals.

Answer 1

for normal distribution z score =(X-μ)/σx
mean μ=	4.1
standard deviation σ=	0.60

a)

probability =P(4<X<4.6)=P((4-4.1)/0.6)<Z<(4.6-4.1)/0.6)=P(-0.17<Z<0.83)=0.7977-0.4338=0.3639

b)

probability =P(X>5)=P(Z>(5-4.1)/0.6)=P(Z>1.5)=1-P(Z<1.5)=1-0.9332=0.0668

c)

for 15th percentile critical value of z=-1.036
therefore corresponding value=mean+z*std deviation=3.5

d)

expected to be longer than 5 minutes =np=225*0.0668 =15.03

e)

here this follows negative binomial distribution with parameter r =8 and p=0.5

P(X=17)=

(x-1Cr-1) pr(1−p)(x-r)    =

0.0873