Question

The mean temperature (of daily maximum temperatures) in July for Dallas–Ft. Worth, Texas, is 85 degrees. Assuming a normal distribution, what would the standard deviation have to be if 10% of days have a high of at least 100 degrees?

Answer 1

Draw a normal curve and shade the region that represents 10% of the right of 100 degrees as shown in Figure (1).

 

Define a random variable (X) as daily maximum temperature with mean (µ) 85 degrees.

 

Draw a standard normal curve and shade the region that represents 0.1 of the right z as shown in Figure (2).

 

From Table E of Cumulative Standard Normal Distribution, the value of z corresponding to 0.90 (=1 – 0.1) is 1.28.

 

Find the standard deviation as shown below:

 

The value of X is found using the formula given below:

X = µ + σz

 

Substitute X as 100, µ as 85, and z as 1.28 in the above Equation

   100 = 85 + (σ)(1.28)

1.28σ = 100 – 85

       σ = 15/1.28

          = 11.7°

 

Thus, the standard deviation is 11.7°.

Thus, the standard deviation is 11.7°.