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Chapter 2: Problem 194

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

Expert verified
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

01

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.
02

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).
03

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.
04

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.
05

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Positive association
In statistics, a positive association exists between two variables when an increase in one variable tends to be linked with an increase in the other variable. In the context of internet connection speeds and time spent online, a positive association would imply that as more individuals have access to faster internet, they also spend more time browsing or performing activities online.

This kind of relationship is quite intuitive. Faster connection speeds mean quicker loading times and smoother streaming experiences. This makes online activities more enjoyable and efficient, therefore encouraging users to stay connected for longer durations.
  • Improved streaming quality
  • Quick downloading of information
  • Enhanced interaction in real-time applications
These benefits all contribute to more extended hours spent online. Essentially, the quicker the connection, the richer the online experience, leading to an increased amount of time users are willing to invest online daily.
Negative association
A negative association between variables suggests that as one variable increases, the other variable tends to decrease. In terms of internet speed and time spent online, this would imply that upgrading to faster connections might reduce the total hours users spend online.

One possible reasoning behind this is that with faster internet, users can perform the same amount of work online in less time. Tasks such as downloading files, streaming videos, or uploading content might be completed so swiftly that users don't need to stay online as long. For example, viewing a high-resolution video might take just a few minutes compared to slower connections that require users to prolong their online time waiting for buffering or downloading completion.
  • Efficient task completion
  • Less idle time
  • Quicker access to needed information
These efficiency improvements could potentially lead to users achieving their needs in a shorter span, thereby reducing overall time spent online.
Scatterplot analysis
Scatterplot analysis is a visual tool used in statistics to examine the relationship between two continuous variables. By plotting data points on a graph with an x and y axis, researchers can observe patterns, trends, and potential correlations between the studied variables.

In the context of the internet speed study, a scatterplot would have "Percent Fast Connection" on the x-axis and "Hours Online" on the y-axis. A glance at the scatterplot can immediately inform us whether there is a positive association, negative association, or no discernible relationship at all.

Points moving in a general upward slope indicate a positive relationship, whereas a downward slope suggests a negative relationship. However, sometimes a scatterplot may reveal points that don't fit the general pattern鈥攖hese are called outliers. In our study, identifying countries with distinct points could help us understand anomalies in internet usage. It's essential to analyze whether these outliers significantly skew the perception of the main correlation by re-plotting the data excluding them. Doing so allows a purer representation of the bulk data, confirming or contesting the initial observed trend.

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Most popular questions from this chapter

Use data on college students collected from the American College Health Association-National College Health Assessment survey \(^{18}\) conducted in Fall 2011 . The survey was administered at 44 colleges and universities representing a broad assortment of types of schools and representing all major regions of the country. At each school, the survey was administered to either all students or a random sample of students, and more than 27,000 students participated in the survey. Students in the ACHA-NCHA survey were asked, "Within the last 12 months, have you been in a relationship (meaning an intimate/coupled/partnered relationship) that was emotionally abusive?" The results are given in Table 2.12 . (a) What percent of all respondents have been in an emotionally abusive relationship? (b) What percent of the people who have been in an emotionally abusive relationship are male? (c) What percent of males have been in an emotionally abusive relationship? (d) What percent of females have been in an emotionally abusive relationship? Table 2.12 Have you been in an emotionally abusive relationship? $$\begin{array}{l|rr|r} \hline & \text { Male } & \text { Female } & \text { Total } \\ \hline \text { No } & 8352 & 16,276 & 24,628 \\ \text { Yes } & 593 & 2034 & 2627 \\ \hline \text { Total } & 8945 & 18,310 & 27,255 \\ \hline\end{array}$$

Two variables are defined, a regression equation is given, and one data point is given. (a) Find the predicted value for the data point and compute the residual. (b) Interpret the slope in context. (c) Interpret the intercept in context, and if the intercept makes no sense in this context, explain why. Weight \(=\) maximum weight capable of bench pressing (pounds), Training = number of hours spent lifting weights a week. Weight \(=95+11.7\) (Training); data point is an individual who trains 5 hours a week and can bench 150 pounds.

Indicate whether the five number summary corresponds most likely to a distribution that is skewed to the left, skewed to the right, or symmetric. (100,110,115,160,220)

Researchers examined all sports-related concussions reported to an emergency room for children ages 5 to 18 in the United States over the course of one year. \({ }^{11}\) Table 2.7 displays the number of concussions in each of the major activity categories. (a) Are these results from a population or a sample? (b) What proportion of concussions came from playing football? (c) What proportion of concussions came from riding bicycles? (d) Can we conclude that, at least in terms of concussions, riding bicycles is more dangerous to children in the US than playing football? Why or why not? $$\begin{array}{l|r}\hline \text { Activity } & \text { Frequency } \\ \hline \text { Bicycles } & 23,405 \\ \text { Football } & 20,293 \\\\\text { Basketball } & 11,507 \\ \text { Playground } & 10,414 \\\\\text { Soccer } & 7,667 \\\\\text { Baseball } & 7,433 \\\\\text { All-Terrain Vehicle } & 5,220 \\\\\text { Hockey } & 4,111 \\\\\text { Skateboarding } & 4,408 \\\\\text { Swimming } & 3,846 \\\\\text { Horseback Riding } & 2,648 \\ \hline \text { Total } & 100,952 \\\\\hline\end{array}$$

In Exercise 1.23, we learned of a study to determine whether just one session of cognitive behavioral therapy can help people with insomnia. In the study, forty people who had been diagnosed with insomnia were randomly divided into two groups of 20 each. People in one group received a one-hour cognitive behavioral therapy session while those in the other group received no treatment. Three months later, 14 of those in the therapy group reported sleep improvements while only 3 people in the other group reported improvements. (a) Create a two-way table of the data. Include totals across and down. (b) How many of the 40 people in the study reported sleep improvement? (c) Of the people receiving the therapy session, what proportion reported sleep improvements? (d) What proportion of people who did not receive therapy reported sleep improvements? (e) If we use \(\hat{p}_{T}\) to denote the proportion from part (c) and use \(\hat{p}_{N}\) to denote the proportion from part (d), calculate the difference in proportion reporting sleep improvements, \(\hat{p}_{T}-\hat{p}_{N}\) between those getting therapy and those not getting therapy.
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