Question

Let's say that the price for a monthly cell phone plan in the US follows a normal distribution with a mean of $62 with a standard deviation of $18. Calculate the z-score for the cost of your cell phone plan. Then write a few complete sentences to describe: Price of my cell phone is $110

• What proportion of cell phone plans are below your cost?
• What proportion of cell phone plans are above your cost?
• Is your cell phone plan considered unusually expensive or inexpensive? (Hint: find the middle 90% of observations

Answer 1

Let X follows a normal distribution with a mean of $62 with a standard deviation of $18.

Here we need to find the z-score for the cost of the cell phone ( x = 110 ) .

The formula of Z score is as follow:

Let's plug the given values in the above formula:

The price of the your cell phone is (8/3) = 2.6667 times higher than the mean price of the cell phones.

a) What proportion of cell phone plans are below your cost?

Here we want to find P( X < 110 ) = P(Z < 2.6667) = "=NORMSDIST(M13)" = 0.9962

Note that here we use excel command to find the normal probability.

b) What proportion of cell phone plans are above your cost?

Here we want to find P( X >110 ) = 1 - P( X < 110 ) = 1 - P(Z < 2.6667) = 1 - 0.9962 = 0.0038

b) Is your cell phone plan considered unusually expensive or inexpensive?

Let's find the range of the middle 90% observations:

such that P( x1 < X < x2) = 0.95

Therefore, P( X < x2 ) = 1 - 0.05 = 0.95

x2 = "=NORMINV(0.95,62,18)" = 91.6

and P( X < x1) = 0.05

x1 = "=NORMINV(0.05,62,18)" = 32.4

Therefore middle 90% prices of the cell phones are lies between $32.4 and $91.6

since $110 is greater than $91.6 (upper limit) , so your cell phone plan considered unusually expensive