Assume the data set described is normally distributed with t

Assume the data set described is normally distributed with the given mean and standard deviation, and with n total values. Find the approximate number of data values that will fall in the given range.

Mean = 110

Standard deviation = 7

n = 210

Range: 96 to 124

In this case, we expect about how many data values to fall between 96 and 124.

Solution

Mean ( u ) =110
Standard Deviation ( sd )=7
Number ( n ) = 210
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 96) = (96-110)/7
= -14/7 = -2
P(X < 124) = (124-110)/7
= 14/7 = 2

Also,
The given range corresponds to those values that lie within 2 standard deviations of the
mean
By empirical rule we know that 95% of values lie within 2 standard deviation
Hence
The number of data value that lie within range: 96 to 124 is (0.95)210 = 199.5~200
Therefore,
200 values lie in the range 96 to 124