Question

In Exercise 2.120 on page \(92,\) we discuss a study in which the Nielsen Company measured connection speeds on home computers in nine different countries in order to determine whether connection speed affects the amount of time consumers spend online. \(^{69}\) Table 2.29 shows the percent of Internet users with a "fast" connection (defined as \(2 \mathrm{Mb}\) or faster) and the average amount of time spent online, defined as total hours connected to the Web from a home computer during the month of February 2011. The data are also available in the dataset GlobalInternet. (a) What would a positive association mean between these two variables? Explain why a positive relationship might make sense in this context. (b) What would a negative association mean between these two variables? Explain why a negative relationship might make sense in this context. $$ \begin{array}{lcc} \hline \text { Country } & \begin{array}{c} \text { Percent Fast } \\ \text { Connection } \end{array} & \begin{array}{l} \text { Hours } \\ \text { Online } \end{array} \\ \hline \text { Switzerland } & 88 & 20.18 \\ \text { United States } & 70 & 26.26 \\ \text { Germany } & 72 & 28.04 \\ \text { Australia } & 64 & 23.02 \\ \text { United Kingdom } & 75 & 28.48 \\ \text { France } & 70 & 27.49 \\ \text { Spain } & 69 & 26.97 \\ \text { Italy } & 64 & 23.59 \\ \text { Brazil } & 21 & 31.58 \\ \hline \end{array} $$ (c) Make a scatterplot of the data, using connection speed as the explanatory variable and time online as the response variable. Is there a positive or negative relationship? Are there any outliers? If so, indicate the country associated with each outlier and describe the characteristics that make it an outlier for the scatterplot. (d) If we eliminate any outliers from the scatterplot, does it appear that the remaining countries have a positive or negative relationship between these two variables? (e) Use technology to compute the correlation. Is the correlation affected by the outliers? (f) Can we conclude that a faster connection speed causes people to spend more time online?

Short answer

A positive association would mean that with a higher internet connection speed, people spend more time online, potentially due to a better user experience. Conversely, a negative association would mean that with a higher internet connection speed, time spent online decreases, potentially because tasks are completed more quickly. After plotting these variables on a scatterplot, any outliers should be identified and their impact on the correlation noted. It cannot be concluded definitively that a faster connection speed causes more time to be spent online, as other factors could influence this.

Step-by-step solution

Understanding positive and negative association

A positive association between the two variables means that as the internet connection speed increases (from a 'fast' connection level), the time consumers spend online also increases. This makes sense as higher connection speeds may offer a better online experience, encouraging users to spend more time online. A negative association, on the other hand, implies that as the internet connection speed increases, the time consumers spend online decreases. This may also make sense because faster connections may allow users to complete their online tasks quicker, thus reducing the total time spent online.

Creating a scatterplot

Plot the percent of internet users with a 'fast' connection (2Mb or faster) on the x-axis and the average amount of time spent online on the y-axis for each country. Identify any countries that lie significantly apart from the main cluster of points as outliers. A positive relationship is indicated if the points show an upward trend (i.e., as the x-values increase, the y-values also increase), and a negative relationship is indicated if the points show a downward trend (i.e., as the x-values increase, the y-values decrease).

Evaluating the effect of outliers

Now, re-plot the scatterplot after removing any outliers identified in the previous step. Gauge whether the relationship between the percentage of fast connection and time spent online changes from being positive or negative.

Calculating the correlation

Use technological tools to calculate the correlation between the percentage of fast connection and the time spent online. Note whether the correlation changes after the removal of the outliers.

Understanding the causality

Although a correlation may exist between connection speed and time spent online, from the data provided, it is not possible to conclude definitively whether a faster connection speed causes people to spend more time online. There could be other factors involved influencing the time spent online, such as the type of online activities preferred in each country.