Question

For homes in a certain state, electric consumption amounts last year approximately followed a mound-shaped (normal) distribution with a mean of 1034 kilowatt-hours and a standard deviation of 182 kilowatt-hours.

(a) According to the empirical rule, approximately 99.7% of values in the distribution will be between these two bounds:
Lower-bound =___ kilowatt-hours and upper-bound = ___ kilowatt-hours.

(b) According to the empirical rule, approximately 68% of values in the distribution will be between these two bounds:
Lower-bound = ___ kilowatt-hours and upper-bound = ___ kilowatt-hours.

(c) According to the empirical rule, approximately 95% of values in the distribution will be between these two bounds:
Lower-bound = ___ kilowatt-hours and upper-bound = ___kilowatt-hours.

Answer 1

Solution :

Given that ,

mean = = 1034 kilowatt-hours

standard deviation = = 182  kilowatt-hours

Using Empirical rule,

a) P( - 3 < x <   + 3 ) = 99.7%

= P( 1034 - 3 * 182 < x < 1034 + 3 * 182 ) = 99.7%

= P( 1034 - 546 < x < 1034 + 546 ) = 99.7%

=P( 488 < x < 1580 ) = 99.7%

Lower-bound = 488 kilowatt-hours and upper-bound = 1580 kilowatt-hours.

b) P( - < x <   + ) = 68%

= P( 1034 - 182 < x < 1034 + 182 ) = 68%

= P( 852 < x < 1216 ) =68%

Lower-bound = 852 kilowatt-hours and upper-bound = 1216 kilowatt-hours.

c) P( - 2 < x <   + 2 ) = 95%

= P( 1034 - 2 * 182 < x < 1034 + 2 * 182 ) = 95%

= P( 1034 - 364 < x < 1034 + 364 ) = 95%

=P( 670 < x < 1398 ) = 95%

Lower-bound = 670 kilowatt-hours and upper-bound = 1398 kilowatt-hours.