Question

It is estimated that amounts of money spent on gasoline by customers at a gas station in Bristol, Englands, follow a normal distribution with a standard deviation of £3,4. It is also found that 5% of all the customers spent more than £30. What percentage of customers spent less than £25? (explain with steps)

Answer 1

Solution:

Given: The amounts of money spent on gasoline by customers at a gas station in Bristol, Englands, follow a normal distribution with a standard deviation of £3.4.

5% of all the customers spent more than £30.

that is:

P( X > 30) = 0.05

We have to find : P( X < 25) =..........?

Find mean of the distribution.

Since P( X > 30) = 0.05

find z value such that:

P( Z> z ) =0.05

that is:

P( Z < z ) = 1 -P( Z> z)

P( Z < z ) = 1 - 0.05

P( Z < z ) = 0.95

Look in z table for Area = 0.9500 or its closest area and find corresponding z value.

Area 0.9500 is in between 0.9495 and 0.9505 and both the area are at same distance from 0.9500

Thus we look for both area and find both z values

Thus Area 0.9495 corresponds to 1.64 and 0.9505 corresponds to 1.65

Thus average of both z values is : ( 1.64+1.65) / 2 = 1.645

Thus z = 1.645

Now use z score formula:

Now use this mean to find P( X < 25)

Find z score for x = 25

thus

P( X < 25) =P( Z< 0.17)

Look in z table for z = 0.1 and 0.07 and find corresponding area.

P( Z < 0.17 )= 0.5675

thus

P( X < 25) =P( Z< 0.17)

P( X < 25) = 0.5675