Question

For a national weather service station in Ohio, the distribution of high temperature readings for the months of July & august is approximately normal, with a mean of 87.5 degrees F, and standard deviation of 6.8 degrees F. What % of days during July & August will have high temperatures:

a. between 87.5 and 95 degrees F

b. above 95 degrees F

c. between 78 and 90 degrees F

d. less than 75 degrees F

Answer 1

Solution:

Given that,

mean = = 87.5

standard deviation = = 6.8

a ) p ( 87.5 < x < 95 )

= p ( 87.5 - 87.5 / 6.8) < ( x -  / ) < ( 95 - 87.5 / 6.8)

= p ( 0 / 6.8 < z < 7.5 / 6.8 )

= p (0 < z < 1.10 )

= p (z < 1.10 ) - p ( z < 0 )

Using z table

= 0.8643 - 0.5000

= 0.3643

Probability = 0.3643

36.43% of days during July & August will have high temperatures.

b ) p ( x > 95 )

= 1 - p (x < 95 )

= 1 - p ( x -  / ) < ( 95 - 87.5 / 6.8)

= 1 - p ( z < 7.5 / 6.8 )

= 1 - p ( z < 1.10)

Using z table

= 1 - 0.8643

= 0.1357

Probability = 0.1357

13.57% of days during July & August will have high temperatures.

c ) p ( 78 < x < 90 )

= p ( 78 - 87.5 / 6.8) < ( x -  / ) < ( 90 - 87.5 / 6.8)

= p ( - 9.5 / 6.8 < z < 2.5 / 6.8 )

= p (-1.40 < z < 0.37 )

= p (z < 0.37 ) - p ( z < - 1.40 )

Using z table

= 0.6443 - 0.0808

= 0.5635

Probability = 0.5635

56.35% of days during July & August will have high temperatures.

d ) p ( x < 75 )

= p ( x -  / ) < ( 75 - 87.5 / 6.8)

= p ( z < - 12.5 / 6.8 )

= p ( z < - 1.84 )

Using z table

= 0.0329

Probability = 0.0329

03.29% of days during July & August will have high temperatures.