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Chapter 9: Problem 12

Do people who spend more time on social networking sites spend more time using Twitter? Megan conducted a study and found that the correlation between the times spent on the two activities was \(0.8\). What does this result say about the relationship between times spent on the two activities? If someone spends more time than average on a social networking site, can you automatically conclude that he or she spends more time than average using Twitter? Explain.

Short Answer

Expert verified
A correlation of 0.8 suggests a strong positive relationship but does not imply causation. More time on social networking likely means more time on Twitter, but it's not certain for every individual.

Step by step solution

01

Understanding Correlation

Correlation is a statistical measure that describes the degree to which two variables move in relation to each other. In this exercise, Megan found the correlation between time spent on social networking sites and time spent on Twitter is 0.8. This value is quite high, suggesting a strong positive relationship between the two activities. This means that, in general, as one spends more time on social networking sites, they tend to spend more time on Twitter as well.
02

Interpreting Correlation Coefficients

A correlation coefficient ranges from -1 to 1. A value of 0 implies no relationship, values closer to 1 indicate a strong positive relationship, and values closer to -1 indicate a strong negative relationship. In this case, the correlation of 0.8 indicates a strong positive association, meaning the two activities often increase together.
03

Understanding What Correlation Does Not Imply

While correlation measures the relationship between two variables, it does not imply causation. A high correlation coefficient, like 0.8, signifies a strong relationship, but it does not automatically mean that spending more time on social networking sites causes more time to be spent on Twitter, or vice versa.
04

Concluding Implications

Given the correlation, if someone spends more time than average on social networking sites, they are likely to spend more time than average using Twitter. However, this is not guaranteed. Individual behavior can vary, and exceptions can exist. This means someone could spend more on other social networking activities or none at all yet still use Twitter a lot less.

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Key Concepts

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

Understanding the Correlation Coefficient
The correlation coefficient is a numerical value that helps us understand the strength and direction of a relationship between two variables. Think of it as a way to see if two things are connected and how strongly. The correlation coefficient ranges from -1 to 1:
  • -1 indicates a perfect negative relationship. As one variable increases, the other decreases.
  • 0 means no discernible relationship between the variables.
  • 1 indicates a perfect positive relationship. As one variable increases, the other also increases.
In Megan's study, a correlation of 0.8 tells us that there is a strong positive relationship. This means that as the time spent on social networking sites goes up, so does the time spent on Twitter. Understanding this number helps in analyzing data patterns, but remember, it doesn鈥檛 explain why these patterns happen.
The Relationship Between Variables
The concept of the relationship between variables is all about how changes in one variable might be linked to changes in another. In Megan's study of social networking and Twitter usage, the observed correlation of 0.8 suggests a noticeable association between the two. However, it is important to understand that a relationship shown by a correlation doesn鈥檛 mean one variable causes the other to change. For example:
  • A strong correlation might occur because both variables are influenced by another hidden factor.
  • One variable might change due to reasons unrelated to the other.
In simple terms, observe the relationship but do not assume a cause-and-effect without further evidence.
Social Media Usage Analytics
Analyzing social media usage can provide insights into how people spend their time online. It's an emerging field that collects data from various platforms like social networking sites and Twitter. In Megan's study, the analytics done using a correlation coefficient of 0.8 shows there鈥檚 a strong link between general social media use and specific Twitter use. Let鈥檚 break down why this matters:
  • Businesses can tailor marketing strategies according to where their audience spends most time.
  • Researchers can explore patterns and trends, offering improved usability or targeted content.
But always remember, while analytics can reflect a trend or correlation, it's only one part of the bigger picture. It shows relationships, not direct causes, meaning findings in social media analytics need careful interpretation to avoid misrepresentation.

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Most popular questions from this chapter

Describe the relationship between two variables when the correlation coefficient \(r\) is (a) near \(-1\). (b) near 0. (c) near 1 .

All Greens is a franchise store that sells house plants and lawn and garden supplies. Although All Greens is a franchise, each store is owned and managed by private individuals. Some friends have asked you to go into business with them to open a new All Greens store in the suburbs of San Diego. The national franchise headquarters sent you the following information at your request. These data are about 27 All Greens stores in California. Each of the 27 stores has been doing very well, and you would like to use the information to help set up your own new store. The variables for which we have data are \(x_{1}=\) annual net sales, in thousands of dollars \(x_{2}=\) number of square feet of floor display in store, in thousands of square feet \(x_{3}=\) value of store inventory, in thousands of dollars \(x_{4}=\) amount spent on local advertising, in thousands of dollars \(x_{5}=\) size of sales district, in thousands of families \(x_{6}=\) number of competing or similar stores in sales district A sales district was defined to be the region within a 5 -mile radius of an All Greens store. $$ \begin{array}{rlrrrr|rrrrrr} \hline x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} \\ \hline 231 & 3 & 294 & 8.2 & 8.2 & 11 & 65 & 1.2 & 168 & 4.7 & 3.3 & 11 \\ 156 & 2.2 & 232 & 6.9 & 4.1 & 12 & 98 & 1.6 & 151 & 4.6 & 2.7 & 10 \\ 10 & 0.5 & 149 & 3 & 4.3 & 15 & 398 & 4.3 & 342 & 5.5 & 16.0 & 4 \\ 519 & 5.5 & 600 & 12 & 16.1 & 1 & 161 & 2.6 & 196 & 7.2 & 6.3 & 13 \\ 437 & 4.4 & 567 & 10.6 & 14.1 & 5 & 397 & 3.8 & 453 & 10.4 & 13.9 & 7 \\ 487 & 4.8 & 571 & 11.8 & 12.7 & 4 & 497 & 5.3 & 518 & 11.5 & 16.3 & 1 \\ 299 & 3.1 & 512 & 8.1 & 10.1 & 10 & 528 & 5.6 & 615 & 12.3 & 16.0 & 0 \\ 195 & 2.5 & 347 & 7.7 & 8.4 & 12 & 99 & 0.8 & 278 & 2.8 & 6.5 & 14 \\ 20 & 1.2 & 212 & 3.3 & 2.1 & 15 & 0.5 & 1.1 & 142 & 3.1 & 1.6 & 12 \\ 68 & 0.6 & 102 & 4.9 & 4.7 & 8 & 347 & 3.6 & 461 & 9.6 & 11.3 & 6 \\ 570 & 5.4 & 788 & 17.4 & 12.3 & 1 & 341 & 3.5 & 382 & 9.8 & 11.5 & 5 \\ 428 & 4.2 & 577 & 10.5 & 14.0 & 7 & 507 & 5.1 & 590 & 12.0 & 15.7 & 0 \\ 464 & 4.7 & 535 & 11.3 & 15.0 & 3 & 400 & 8.6 & 517 & 7.0 & 12.0 & 8 \\ 15 & 0.6 & 163 & 2.5 & 2.5 & 14 & & & & & & \\ \hline \end{array} $$ (a) Generate summary statistics, including the mean and standard deviation of each variable. Compute the coefficient of variation (see Section \(3.2\) ) for each variable. Relative to its mean, which variable has the largest spread of data values? Which variable has the least spread of data values relative to its mean? (b) For each pair of variables, generate the sample correlation coefficient \(r .\) For all pairs involving \(x_{1}\), compute the corresponding coefficient of determination \(r^{2}\). Which variable has the greatest influence on annual net sales? Which variable has the least influence on annual net sales? (c) Perform a regression analysis with \(x_{1}\) as the response variable. Use \(x_{2}, x_{3}\), \(x_{4}, x_{5}\), and \(x_{6}\) as explanatory variables. Look at the coefficient of multiple determination. What percentage of the variation in \(x_{1}\) can be explained by the corresponding variations in \(x_{2}, x_{3}, x_{4}, x_{5}\), and \(x_{6}\) taken together? (d) Write out the regression equation. If two new competing stores moved into the sales district but the other explanatory variables did not change, what would you expect for the corresponding change in annual net sales? Explain your answer. If you increased the local advertising by a thousand dollars but the other explanatory variables did not change, what would you expect for the corresponding change in annual net sales? Explain. (e) Test each coefficient to determine if it is or is not zero. Use level of significance \(5 \%\). (f) Suppose you and your business associates rent a store, get a bank loan to start up your business, and do a little research on the size of your sales district and the number of competing stores in the district. If \(x_{2}=2.8\), \(x_{3}=250, x_{4}=3.1, x_{5}=7.3\), and \(x_{6}=2\), use a computer to forecast \(x_{1}=\) annual net sales and find an \(80 \%\) confidence interval for your forecast (if your software produces prediction intervals). (g) Construct a new regression model with \(x_{4}\) as the response variable and \(x_{1}\), \(x_{2}, x_{3}, x_{5}\), and \(x_{6}\) as explanatory variables. Suppose an All Greens store in Sonoma, California, wants to estimate a range of advertising costs appropriate to its store. If it spends too little on advertising, it will not reach enough customers. However, it does not want to overspend on advertising for this type and size of store. At this store, \(x_{1}=163, x_{2}=2.4, x_{3}=188\), \(x_{5}=6.6\), and \(x_{6}=10\). Use these data to predict \(x_{4}\) (advertising costs) and find an \(80 \%\) confidence interval for your prediction. At the \(80 \%\) confidence level, what range of advertising costs do you think is appropriate for this store?

Over the past 30 years in the United States, there has been a strong negative correlation between the number of infant deaths at birth and the number of people over age 65 . (a) Is the fact that people are living longer causing a decrease in infant mortalities at birth? (b) What lurking variables might be causing the increase in one or both of the variables? Explain.

Over the past decade, there has been a strong positive correlation between teacher salaries and prescription drug costs. (a) Do you think paying teachers more causes prescription drugs to cost more? Explain. (b) What lurking variables might be causing the increase in one or both of the variables? Explain.

Suppose you are interested in buying a new Lincoln Navigator or Town Car. You are standing on the sales lot looking at a model with different options. The list price is on the vehicle. As a salesperson approaches, you wonder what the dealer invoice price is for this model with its options. The following data are based on information taken from Consumer Guide (Vol. 677). Let \(x\) be the list price (in thousands of dollars) for a random selection of these cars of different models and options. Let \(y\) be the dealer invoice (in thousands of dollars) for the given vehicle. $$ \begin{array}{l|lllll} \hline x & 32.1 & 33.5 & 36.1 & 44.0 & 47.8 \\ \hline y & 29.8 & 31.1 & 32.0 & 42.1 & 42.2 \\ \hline \end{array} $$ (a) Verify that \(\Sigma x=193.5, \quad \Sigma y=177.2, \quad \Sigma x^{2}=7676.71, \quad \Sigma y^{2}=6432.5\), \(\Sigma x y=7023.19\), and \(r \approx 0.977\) (b) Use a \(1 \%\) level of significance to test the claim that \(\rho>0\). (c) Verify that \(S_{e} \approx 1.5223, a \approx 1.4084\), and \(b \approx 0.8794\). (d) Find the predicted dealer invoice when the list price is \(x=40\) (thousand dollars). (e) Find a \(95 \%\) confidence interval for \(y\) when \(x=40\) (thousand dollars). (f) Use a \(1 \%\) level of significance to test the claim that \(\beta>0\). (g) Find a \(90 \%\) confidence interval for \(\beta\) and its meaning.
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