Question

1. The Energy information Administration reported that the mean retail price per gallon of regular grade gasoline was $3.43 (Energy Information Administration, July 2012). Suppose that the standard deviation was $0.10 and that the retail price per gallon has a bell-shaped distribution.

1. What percentage of regular grade gasoline sold between $3.33 and $3.53 per gallon?
   
2. What percentage of regular grade gasoline sold between $3.33 and $3.63 per gallon?

c. What percentage of regular grade gasoline sold for more than $3.63 per gallon?

Answer 1

The given information is that the data have bell shaped distribution with a mean of $ 3.43 and a standard deviation of $ 0.10.

The formula for z score is,

Z = x - Mean / SD

Where,

x is the retail price per gallon.

Mean is the sample mean and

SD is the standard deviation.

a) The z - score corresponding to $ 3.33 and $ 3.53 can be obtained as follows.

Z = (3.33 - 3.43) / 0.10

= -0.1 / 0.10

= -1.

Similarly for $ 3.53

Z = (3.53 - 3.43) / 0.10

= 0.1 / 0.10

= 1.

We need to calculate percentage of regular grade gasoline sold between $3.33 and $3.53 per gallon.

Percentage of regular grade gasoline sold between $3.33 and $3.53 per gallon.

= Percentage area for (-1 < z < 1).

= Percentage area for (z < 1) - Percentage area for ( z < -1).

= Percentage for (0.8413) - Percentage for (0.1587)

= 84.13% - 15.87%

= 68.26%

Conclusion : The percentage of regular grade gasoline sold between $3.33 and $3.53 per gallon is 68.26%.

b) The z - score corresponding to $ 3.33 and $ 3.63 can be obtained as follows.

Z = (3.33 - 3.43) / 0.10

= -0.1 / 0.10

= -1.

Similarly for $ 3.63

Z = (3.63 - 3.43) / 0.10

= 0.2 / 0.10

= 2.

We need to calculate percentage of regular grade gasoline sold between $3.33 and $3.63 per gallon.

Percentage of regular grade gasoline sold between $3.33 and $3.63 per gallon.

= Percentage area for (-1 < z < 2).

= Percentage area for (z < 2) - Percentage area for ( z < -1).

= Percentage for (0.9772) - Percentage for (0.1587)

= 97.72% - 15.87%

= 81.85%

Conclusion : The percentage of regular grade gasoline sold between $3.33 and $3.63 per gallon is 81.85%.

c )

The z - score corresponding to $ 3.63 can be obtained as follows.

Z = (3.63 - 3.43) / 0.10

= 0.2 / 0.10

= 2.

We need to calculate percentage of regular grade gasoline sold for more than $3.63 per gallon.

Percentage of regular grade gasoline sold for more than $3.63 per gallon is

= Percentage area for ( z > 2)

But z is the standard normal score and z table do not give greater than area so here we need to write it in terms of  

= 100 - percentage area for ( z < 2)

= 100 - percentage for (0.9772 )

= 100 - 97.72%

= 2.28%

Conclusion : The percentage of regular grade gasoline sold more than $3.63 per gallon is 2.28%.