EXAMPLE 5.28

Let $X=$ the number of different people sent text messages during a particular day by a randomly selected student at a large university. Suppose the mean value of $X$ is 7 and the standard deviation is 6 (values very close to those reported in the article “Cell Phone Use and Grade Point Average Among Undergraduate University Students” (College Student J., 2011: 544-551). Among 100 randomly selected such students, how likely is it that the sample mean number of different people texted exceeds 5 ? Notice that the distribution being sampled is discrete, but the CLT is applicable whether the variable of interest is discrete or continuous. Also, although the fact that the standard deviation of this nonnegative variable is quite large relative to the mean value suggests that its distribution is positively skewed, the large sample size implies that $\bar{X}$ does have approximately a normal distribution. Using ${\mu }_{\bar{X}}=7$ and ${\sigma }_{\bar{X}}=.6$ ,

$$P\left(\bar{X}>5\right)\approx P\left(Z>\frac{5-7}{.6}\right)=1-\Phi \left(-3.33\right)=.9996$$

Note: The cited article stated that text messaging frequency was negatively correlated with GPA.