4 30 points Nurnaz uses her mobile phone for X minutes each

4. (30 points) Nurnaz uses her mobile phone for X minutes each day. X is a random variable that can be modeled by a normal distribution with mean 27 minutes and standard deviation of 7 minutes. a. Find the probability that on a particular day Nurnaz uses her mobile phone for i. more than 20,o2 minutes; ii. between 10 and 20 minutes b. Find the probability that on S randomly selected days the mean time Nurnaz talks on her phone is at least 30 minutes

Solution

Normal Distribution
Mean ( u ) =27
Standard Deviation ( sd )=7
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
i)
P(X > 20) = (20-27)/7
= -7/7 = -1
= P ( Z >-1) From Standard Normal Table
= 0.8413                  
ii)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 10) = (10-27)/7
= -17/7 = -2.4286
= P ( Z <-2.4286) From Standard Normal Table
= 0.00758
P(X < 20) = (20-27)/7
= -7/7 = -1
= P ( Z <-1) From Standard Normal Table
= 0.15866
P(10 < X < 20) = 0.15866-0.00758 = 0.1511                  

b)
Normal Distribution
Mean ( u ) =27
Standard Deviation ( sd )=7
Number ( n ) = 5
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  

P(X < 30) =   (30-27)/7/ Sqrt ( 5 )
= 3/3.1305= 0.9583
= P ( Z <0.9583) From Standard NOrmal Table
= 0.831                  
P(X > = 30) = 1 - P(X < 30)
= 1 - 0.831 = 0.169