The average daily temperature in October in Boston is 61F with a standard deviat

Question

The average daily temperature in October in Boston is 61F with a standard deviation of 5 F. These temperatures closely follow a normal distribution.

1. What is the probability of observing a 67F temperature or higher in Boston during a randomly chosen day in October?

2. How warm are the hottest 20% of the days (days with highest average high temperature) during October in Boston?

We use the following equation to convert F (Fahrenheit) to C (Celsius): C = 5 9 (F 32).

4. Write the probability model for the distribution of temperature in C in October in Boston.

5. What is the probability of observing a 20C (which roughly corresponds to 68F) temperature or higher in October in Boston? Calculate using the C model from question 4.

6. Estimate the IQR of the temperatures (in C) in October in Boston.

Explanation / Answer

(1) = 61, = 5

z = (67 - 61)/5 = 1.2

P(x > 67) = P(z > 1.2) = 0.1151

(2) z = 0.8416

x = + z * = 61 + 0.8416 * 5 = 65.21

(4) = (5/9)(61 - 32) = 16.11, = -15

(5) z = (20 - 16.11)/-15 = -0.2593

P(x > 20) = P(z > -0.2593) = 0.6023

(6) Q3 = 16.11 + 0.6745 * -15 = 5.9925 and Q1 = 16.11 - 0.6745 * -15 = 26.23

IQR = 26.23 - 5.9925 = 20.24

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