Question

Temperatures in LA. The average daily high temperature in March in Los Angeles is 70 degrees Fahrenheit with a standard deviation of 5 degrees Fahrenheit. Suppose that the temperatures in March closely follow a normal distribution. Round calculated answers to 4 decimal places. 1. What is the probability of observing an 75 degrees Fahrenheit temperature or higher in Los Angeles during a randomly chosen day in March? .1586 2. How cold are the coldest 5% of the days during March in Los Angeles? or less. We use the following equation to convert degrees Fahrenheit to degrees Celsius: C=(F−32)∗5/9 3. Write the probability model for the distribution of temperature in degrees Celsius in month in Los Angeles. Note: To calculate the new standard deviation, just multiply by 5/9, do not subtract 32. N( , ) 4. What is the probability of observing a 23.889 degrees Celsius (which roughly corresponds to 75 degrees Fahrenheit) temperature or higher in March in Los Angeles? Calculate using the degrees Celsius model from part (3).

Answer 1

X = temperature in Fahrenheit

X = N(70,5^2)

Z= (X - mena)/sd = (X - 70)/5

1)

P(X > 75)

= P(Z > (75 - 70)/5)

= P(Z > 1)

= 1 - P(Z < 1)

= 1- normsdist(1)   {using excel or using normal tables}

= 0.1587

2)

P(X < x) = 0.05

P(Z < z) = 0.05

z= -1.645    { =NORMSINV(0.05)}

X = mena + z * sd

= 70 -1.645 * 5

= 61.775

3)

C = 5/9 (F - 32)

=5/9 F - 32*5/9

if X follow N( , )

then

aX + b follow N(a + b , a^2 )

hence

C follow N( 5/9 * 70 - 32*5/9 , (5/9 * 5)^2)

= N( 21.1111 , 2.7778^2)

4)

P(C > 23.889)

= P(Z > (23.889 - 21.111)/2.7778)

= P(Z > 1)

= 0.1587   {same as 1)

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