Question

Given a normal distribution with μ=103 and σ=25​, and given you select a sample of n=25​, complete parts​ (a) through​ (d).

What is the probability that X is between 91 and 93.5​?

​P(91<X<93.5​)=

Answer 1

The most popular theoretical probability distribution used to deal with continuous variables is normal distribution.

Total area under a normal curve represents total probability. As normal distribution is a probability density function, total probability is equal to unity. Hence, the total area under the normal curve is taken as '1'.

Important properties of a normal distribution

1. Normal curve is bell shaped.

2. All odd central moments of the distribution are equal to zero.

3. Mean = median = mode.

4. It is a uni - modal distribution.

are the two parameters of normal distribution. For different data, values of mu (arithmetic mean)and sigma (standard deviation)will be different.

A standardised form of normal distribution by taking mu = 0 and sigma = 1 is called standard normal distribution. While, the normal variable is usually denoted as 'x', the standard normal variable is denoted as 'z'.

A normal curve with arithmetic mean = and standard deviation = can be converted into a standard normal curve by changing both origin and scale.

The formula used to convert the normal variable, 'x' into the standard normal variable, 'z' is given as ,

z =( x- ) /  

In the given question, = 103, = 25.

P(91< x < 93.5) =?

Let , X = 91, Let, X = 93.5,

Z = ( x- ) /     Z = ( x- ) /  

=( 91 - 103) / 25 = (93.5 - 103) / 25

= - 12 /25 = -9.5 /25

= -.48 = -.38

P(z < -.48) = .1844 P(z <-.38) = .1480

ie; P(-.48 < X < -.38 ) =   .1844 - .1480

= .0364

= 3.64 %