Question

Downloading Apps for Your Smartphone A random sample of \(n=461\) smartphone users in the US in January 2015 found that 355 of them have downloaded an app. \(^{10}\) (a) Give notation for the parameter of interest, and define the parameter in this context. (b) Give notation for the quantity that gives the best estimate and give its value. (c) What would we have to do to calculate the parameter exactly?

Short answer

The parameter of interest, denoted as \(P\), represents the proportion of all smartphone users in the US who download an app. The best estimate of this parameter is the sample proportion, denoted as \(\hat{p}\), which in this case is approximately 0.77 or 77%. To calculate the exact value of \(P\), we would need data on every smartphone user in the US, not just a sample.

Step-by-step solution

Identify the parameter of interest and its notation

A parameter of interest is a characteristic of the total population that we want to study. In this context, the parameter of interest is the true proportion of all smartphone users in the US who download apps. We denote this in statistical notation as \(P\).

Identify the estimate of the parameter and its notation

The quantity that gives the best estimate of the parameter is the sample proportion. In statistical notation, we often denote the sample proportion as \(\hat{p}\). The value of \(\hat{p}\) in this case is computed as the number of successes (people who have downloaded apps) divided by the total sample size. Here, that would be \( \hat{p} = \frac{355}{461}\). Computing this value, we get \(\hat{p} \approx 0.77\) or 77%.

Explain the process to calculate the exact parameter

To calculate the exact parameter (P), we would have to devote substantial resources to gather data on every single smartphone user in the U.S, not just a random sample. This would involve collecting data on whether or not each user has downloaded an app. Once the data is collected, the parameter can be calculated by dividing the number of successes by the total population size.