Question

The average daily high temperature in June in LA is 78∘F with a standard deviation of 8∘F. Suppose that the temperatures in June closely follow a normal distribution.

(a) What is the probability of observing a 70∘F temperature or higher in LA during a randomly chosen day in June?
(b) What is the probability of observing a 71∘F temperature or lower in LA during a randomly chosen day in June?
(c) What is the probability of observing a temperature between 79∘F and 85∘F in LA during a randomly chosen day in June?
(d) Find a temperature so that 20% of days will be warmer during June in LA. ∘F

Answer 1

Mean, = 78°F

Standard deviation, = 8°F

Let X (in °F) denote the observed temperature in LA during a randomly chosen day in June

(a) The required probability = P(X ≥ 70)

= P{Z ≥ (70 - 78)/8}

= P{Z ≥ -1) = 0.8413

(b) The required probability = P(X ≤ 71)

= P{Z ≤ (71 - 78)/8}

= P(Z ≤ -0.875)

= 0.1908

(c) The required probability = P(79 ≤ X ≤ 85)

= P{(79 - 78)/8 ≤ Z ≤ (85 - 78)/8}

= P(0.125 ≤ Z ≤ 0.875)

= 0.2594

(d) Corresponding to the top 20% warm days, the critical z score = 0.8418

Thus, the corresponding temperature = = 84.73°F