According to a study among adults in the 25 to 34year age gr

According to a study among adults in the 25- to 34-year age group, the mean amount spent per year on reading and entertainment is $1,994. Assume that the distribution of the amounts spent follows the normal distribution with a standard deviation of $450. What percent of the adults spend more than $2,500 per year on reading and entertainment? What percent spend between $2,500 and $3,000 per year on reading and entertainment? What percent spend less than $1,000 per year on reading and entertainment?

Solution

Normal Distribution
Mean ( u ) =1994
Standard Deviation ( sd )=450
Normal Distribution = Z= X- u / sd ~ N(0,1)                                      
a)
P(X > 2500) = (2500-1994)/450
= 506/450 = 1.1244
= P ( Z >1.124) From Standard Normal Table
= 0.1304                  
b)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 2500) = (2500-1994)/450
= 506/450 = 1.1244
= P ( Z <1.1244) From Standard Normal Table
= 0.86959
P(X < 3000) = (3000-1994)/450
= 1006/450 = 2.2356
= P ( Z <2.2356) From Standard Normal Table
= 0.98731
P(2500 < X < 3000) = 0.98731-0.86959 = 0.1177                  
c)
P(X < 1000) = (1000-1994)/450
= -994/450= -2.2089
= P ( Z <-2.2089) From Standard Normal Table
= 0.0136