Question

The speed at which you can log into a website through a smartphone is an important quality characteristic of that website. In a recent test, the mean time to log into the JetBlue Airways website through a smartphone was 4.237 seconds. Suppose that the download time is normally distributed, with a standard deviation of 1.3 seconds. What is the probability that the download time is:

A. Less than 2 seconds?

B. Between 1.5 and 2.5 seconds

C. Above 1.8 seconds

D. 99% of the download times are slower (higher) than how many seconds?

E. 95% of the download times are between what two values, symmetrically distributed around the mean?

F. Suppose that the download times are uniformly distributed between 1 and 9 seconds. What are your answers to A through C?

Answer 1

A.) 0.0426

B) 0.073121

C.) 0.96958

D.) 1.2 seconds

E.) 1.689 < x < 6.785

Explanation:

Given a normal distribution :

Mean download time (m) = 4.237 seconds

Standard deviation (s) = 1.3 seconds

A.) Probability that download time is less than 2 seconds :

P(x < 2)

Obtain standardized score (Z) :

Z = (x - m) / s

Z = (2 - 4.237) / 1.3

Z = −1.720769

P(Z < - 1.7208) = 0.0426 (Z probability calculator)

B.) Between 1.5 and 2.5 seconds :

1.5 < z < 2.5

((x - m) / s) < z <((x - m) / s)

((1.5 - 4.237) / 1.3) < z < ((2.5 - 4.237) / 1.3)

−2.105384 < z < −1.336153

P(Z < - 2.10538) = 0.017629

P(Z −1.336153) = 0.09075

0.09075 - 0.017629 = 0.073121

C.) Above 1.8 seconds

P(x > 1.8)

Obtain standardized score (Z) :

Z = (x - m) / s

Z = (1.8 - 4.237) / 1.3

Z = −1.874615

P(Z > −1.874615) = 0.96958 (Z probability calculator)

D.) 99% of the download times are slower (higher) than how many seconds?

P(Z > x) = 99% = 0.99

Z value corresponds to -2.326

Z = (x - m) / s

-2.326 = (x - 4.237) / 1.3

-2.326 * 1.3 = x - 4.237

−3.0238 = x - 4.237

−3.0238 + 4.237 = x

x = 1.2132

x = 1.2 seconds

E. 95% of the download times are between what two values, symmetrically distributed around the mean?

For symmetric distribution around the mean ;

P(-Z<x<Z) = 0.95

Corresponds to a z value of 1.96

-1.96 * 1.3 = x - 4.237

-2.548 = x - 4.237

x = - 2.548 + 4.237

x = 1.689

1.96 * 1.3 = x - 4.237

2.548 = x - 4.237

x = 2.548 + 4.237

x = 6.785

1.689 < x < 6.785