Question

Downloading “apps” to your cell phone. Refer toExercise 4.173 (p. 282) and the August 2011 survey by thePew Internet & American Life Project. The study foundthat 40% of adult cell phone owners have downloadedan application (“app”) to their cell phone. Assume thispercentage applies to the population of all adult cell phoneowners.

1. In a random sample of 50 adult cell phone owners, howlikely is it to find that more than 60% have downloadedan “app” to their cell phone?
2. Refer to part a. Suppose you observe a sample proportionof .62. What inference can you make about the trueproportion of adult cell phone owners who have downloadedan “app”?
3. Suppose the sample of 50 cell phone owners is obtainedat a convention for the International Association forthe Wireless Telecommunications Industry. How willyour answer to part b change, if at all?

Short answer

1. There is 0.2% chance that more than 60% have downloaded the application.
2. The sample proportion is too small. So, there can assume that either the sample is taken randomly or there is a problem in the population mean proportion.

c. The population mean can’t be rejected and there is no problem with that. So, the sample is taken randomly

Step-by-step solution

Given information

Referring to exercise 4.173 (p. 282), the study discovers that the application was downloaded by 40% of adults on their cell phones. The percentage applied to the population of adult cell phone owners.

So, the probability$p=0.40$

Calculate the Probability

a.

Consider the standard error of the 50 random sample,$\sigma$

So,

$\begin{array}{rcl}\sigma & =& \sqrt{\frac{p\left(1-p\right)}{n}}\\ & =& \sqrt{\frac{0.40\times 0.60}{50}}\\ & =& 0.692\\ & & \end{array}$

Now, the probability that more than 60% have downloaded the application is,

$\begin{array}{rcl}\mathrm{Pr}\left(X>0.60\right)& =& \mathrm{Pr}\left(\frac{\hat{p}-p}{\sigma }>\frac{0.60-0.40}{0.692}\right)\\ & =& \mathrm{Pr}\left(z>2.89\right)\\ & =& 1-\mathrm{Pr}\left(z<2.89\right)\\ & =& 1-0.99807\\ & =& 0.0020\\ & & \end{array}$

Therefore, there is a very low chance of 0.2% that more than 60% have downloaded the application.

Determine the inference

b.

Referring to part a., the sample proportion is too small. So, there can assume that either the sample is taken randomly or there is a problem in the population mean proportion.

Justify the inference

c.

Hence the sample is from the International Association for the Wireless Telecommunication industry. So, the sample is biased and categorizes the adults from the same industry who are more interested to download the application.

Therefore, the population mean can’t be rejected and there is no problem with that. So, the sample is taken randomly.