Question

The mean number of text messages sent per month by customers of a cell phone service provider is 1,650 , and the standard deviation is \(750 .\) Find the \(z\) -score associated with each of the following numbers of text messages sent. a. 0 b. \(\quad 10,000\) c. \(\quad 4,500\) d. \(\quad 300\)

Short answer

The z-scores for 0, 10,000, 4,500 and 300 text messages sent are -2.2, 11.13, 3.8 and -1.8 respectively.

Step-by-step solution

Understand the z-score formula

The z-score formula is given as \(z = \frac{x - \mu}{\sigma}\), where \(x\) is the value for which we want to find the z-score, \(\mu\) is the mean and \(\sigma\) is the standard deviation.

Calculate z-score for 0 messages

Substitute \(x = 0\), \(\mu = 1650\) and \(\sigma = 750\) in the z-score formula. The z-score, \(z = \frac{0 - 1650}{750} = -2.2\).

Calculate z-score for 10,000 messages

Substitute \(x = 10000\), \(\mu = 1650\) and \(\sigma = 750\) in the z-score formula. The z-score, \(z = \frac{10000 - 1650}{750} = 11.13\).

Calculate z-score for 4,500 messages

Substitute \(x = 4500\), \(\mu = 1650\) and \(\sigma = 750\) in the z-score formula. The z-score, \(z = \frac{4500 - 1650}{750} = 3.8\).

Calculate z-score for 300 messages

Substitute \(x = 300\), \(\mu = 1650\) and \(\sigma = 750\) in the z-score formula. The z-score, \(z = \frac{300 - 1650}{750} = -1.8\).